J K displaystyle mathrm tfrac J K , or kilogram metre squared per kelvin second squared k g ⋅ m 2 K ⋅ s 2 displaystyle mathrm tfrac kgcdot m^ 2 Kcdot s^ 2 in the
Contents 1 History 2 Units 2.1 Extensive properties 2.2 Intensive properties 2.3 Alternative unit systems 3 Measurement 3.1 Calculation from first principles 3.2 Thermodynamic relations and definition of heat capacity 3.3 Relation between heat capacities 3.3.1 Ideal gas 3.4 Specific heat capacity
3.5
4 Theory 4.1 Factors that affect specific heat capacity 4.1.1 Degrees of freedom 4.1.2 Example of temperature-dependent specific heat capacity, in a diatomic gas 4.1.3 Per mole of different units 4.1.3.1 Per mole of molecules 4.1.3.2 Per mole of atoms 4.1.4 Corollaries of these considerations for solids (volume-specific heat capacity) 4.1.5 Other factors 4.1.5.1
4.2 The simple case of the monatomic gas
4.3
5 Table of specific heat capacities
6
History[edit]
Main article: History of heat
In a previous theory of heat common in the early modern period, heat
was thought to be a measurement of an invisible fluid, known as the
caloric. Bodies were capable of holding a certain amount of this
fluid, hence the term heat capacity, named and first investigated by
Scottish chemist
C ( T ) = δ Q d T , displaystyle C(T)= frac delta Q dT , where the symbol δ designates heat as a path function. If the temperature change is sufficiently small the heat capacity may be assumed to be constant: C = Q Δ T . displaystyle C= frac Q Delta T .
Q displaystyle Q ) to achieve the same change in temperature ( Δ T displaystyle Delta T ). Intensive properties[edit] For many purposes it is more convenient to report heat capacity as an intensive property, an intrinsic characteristic of a particular substance. In practice, this is most often an expression of the property in relation to a unit of mass; in science and engineering, such properties are often prefixed with the term specific.[6] International standards now recommend that specific heat capacity always refer to division by mass.[7] The units for the specific heat capacity are [ c ] = J k g ⋅ K displaystyle [c]=mathrm tfrac J kgcdot K . In chemistry, heat capacity is often specified relative to one mole, the unit of amount of substance, and is called the molar heat capacity. It has the unit [ C m o l ] = J m o l ⋅ K displaystyle [C_ mathrm mol ]=mathrm tfrac J molcdot K . For some considerations it is useful to specify the volume-specific heat capacity, commonly called volumetric heat capacity, which is the heat capacity per unit volume and has SI units [ s ] = J m 3 ⋅ K displaystyle [s]=mathrm tfrac J m^ 3 cdot K . This is used almost exclusively for liquids and solids, since for
gases it may be confused with specific heat capacity at constant
volume.
Alternative unit systems[edit]
While SI units are the most widely used, some countries and industries
also use other systems of measurement. One older unit of heat is the
kilogram-calorie (Cal), originally defined as the energy required to
raise the temperature of one kilogram of water by one degree Celsius,
typically from 14.5 to 15.5 °C. The specific average heat
capacity of water on this scale would therefore be exactly
1 Cal/(C°⋅kg). However, due to the temperature-dependence of
the specific heat, a large number of different definitions of the
calorie came into being. Whilst once it was very prevalent, especially
its smaller cgs variant the gram-calorie (cal), defined thus the
specific heat of water would be 1 cal/(K⋅g), in most fields the
use of the calorie is now archaic.
In the United States other units of measure for heat capacity may be
quoted in disciplines such as construction, civil engineering, and
chemical engineering. A still common system is the English Engineering
Units in which the mass reference is pound mass and the temperature is
specified in degrees
For liquids and gases, it is important to know the pressure to which
given heat capacity data refer. Most published data are given for
standard pressure. However, quite different standard conditions for
temperature and pressure have been defined by different organizations.
The
Δ e system = e in − e out , displaystyle Delta e_ text system =e_ text in -e_ text out , or d U = δ Q + δ W . displaystyle mathrm d U=delta Q+delta W. For work as a result of an increase of the system volume we may write d U = δ Q − P d V . displaystyle mathrm d U=delta Q-P,mathrm d V. If the heat is added at constant volume, then the second term of this relation vanishes, and one readily obtains ( ∂ U ∂ T ) V = ( ∂ Q ∂ T ) V = C V . displaystyle left( frac partial U partial T right)_ V =left( frac partial Q partial T right)_ V =C_ V . This defines the heat capacity at constant volume, CV, which is also related to changes in internal energy. Another useful quantity is the heat capacity at constant pressure, CP. This quantity refers to the change in the enthalpy of the system, which is given by H = U + P V . displaystyle H=U+PV. A small change in the enthalpy can be expressed as d H = δ Q + V d P , displaystyle mathrm d H=delta Q+V,mathrm d P, and therefore, at constant pressure, we have ( ∂ H ∂ T ) P = ( ∂ Q ∂ T ) P = C P . displaystyle left( frac partial H partial T right)_ P =left( frac partial Q partial T right)_ P =C_ P . These two equations: ( ∂ U ∂ T ) V = ( ∂ Q ∂ T ) V = C V , displaystyle left( frac partial U partial T right)_ V =left( frac partial Q partial T right)_ V =C_ V , ( ∂ H ∂ T ) P = ( ∂ Q ∂ T ) P = C P displaystyle left( frac partial H partial T right)_ P =left( frac partial Q partial T right)_ P =C_ P are property relations and are therefore independent of the type of process. In other words, they are valid for any substance going through any process. Both the internal energy and enthalpy of a substance can change with the transfer of energy in many forms i.e., heat.[10] Relation between heat capacities[edit] Main article: Relations between heat capacities Measuring the heat capacity, sometimes referred to as specific heat, at constant volume can be prohibitively difficult for liquids and solids. That is, small temperature changes typically require large pressures to maintain a liquid or solid at constant volume, implying that the containing vessel must be nearly rigid or at least very strong (see coefficient of thermal expansion and compressibility). Instead, it is easier to measure the heat capacity at constant pressure (allowing the material to expand or contract freely) and solve for the heat capacity at constant volume using mathematical relationships derived from the basic thermodynamic laws. Starting from the fundamental thermodynamic relation one can show that C P − C V = T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , displaystyle C_ P -C_ V =Tleft( frac partial P partial T right)_ V,n left( frac partial V partial T right)_ P,n , where the partial derivatives are taken at constant volume and constant number of particles, and constant pressure and constant number of particles, respectively. This can also be rewritten as C P − C V = V T α 2 β T , displaystyle C_ P -C_ V =VT frac alpha ^ 2 beta _ T , where α displaystyle alpha is the coefficient of thermal expansion, β T displaystyle beta _ T is the isothermal compressibility. The heat capacity ratio, or adiabatic index, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor. Ideal gas[edit] [11] For an ideal gas, evaluating the partial derivatives above according to the equation of state, where R is the gas constant, for an ideal gas P V = n R T , displaystyle PV=nRT, C P − C V = T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n , displaystyle C_ P -C_ V =Tleft( frac partial P partial T right)_ V,n left( frac partial V partial T right)_ P,n , P = n R T V ⇒ ( ∂ P ∂ T ) V , n = n R V , displaystyle P= frac nRT V Rightarrow left( frac partial P partial T right)_ V,n = frac nR V , V = n R T P ⇒ ( ∂ V ∂ T ) P , n = n R P . displaystyle V= frac nRT P Rightarrow left( frac partial V partial T right)_ P,n = frac nR P . Substituting T ( ∂ P ∂ T ) V , n ( ∂ V ∂ T ) P , n = T n R V n R P = n R T V n R P = P n R P = n R , displaystyle Tleft( frac partial P partial T right)_ V,n left( frac partial V partial T right)_ P,n =T frac nR V frac nR P = frac nRT V frac nR P =P frac nR P =nR, this equation reduces simply to Mayer's relation: C P , m − C V , m = R . displaystyle C_ P,m -C_ V,m =R. The differences in heat capacities as defined by the above Mayer relation is only exact for an ideal gas and would be different for any real gas. Specific heat capacity[edit] The specific heat capacity of a material on a per mass basis is c = ∂ C ∂ m , displaystyle c= frac partial C partial m , which in the absence of phase transitions is equivalent to c = E m = C m = C ρ V , displaystyle c=E_ m = frac C m = frac C rho V , where C displaystyle C is the heat capacity of a body made of the material in question, m displaystyle m is the mass of the body, V displaystyle V is the volume of the body, ρ = m V displaystyle rho = frac m V is the density of the material. For gases, and also for other materials under high pressures, there is need to distinguish between different boundary conditions for the processes under consideration (since values differ significantly between different conditions). Typical processes for which a heat capacity may be defined include isobaric (constant pressure, d P = 0 displaystyle text d P=0 ) or isochoric (constant volume, d V = 0 displaystyle text d V=0 ) processes. The corresponding specific heat capacities are expressed as c P = ( ∂ C ∂ m ) P , displaystyle c_ P =left( frac partial C partial m right)_ P , c V = ( ∂ C ∂ m ) V . displaystyle c_ V =left( frac partial C partial m right)_ V . From the results of the previous section, dividing through by the mass gives the relation c P − c V = α 2 T ρ β T . displaystyle c_ P -c_ V = frac alpha ^ 2 T rho beta _ T . A related parameter to c displaystyle c is C / V displaystyle C/V , the volumetric heat capacity. In engineering practice, c V displaystyle c_ V for solids or liquids often signifies a volumetric heat capacity, rather than a constant-volume one. In such cases, the mass-specific heat capacity (specific heat) is often explicitly written with the subscript m displaystyle m , as c m displaystyle c_ m . Of course, from the above relationships, for solids one writes c m = C m = c volumetric ρ . displaystyle c_ m = frac C m = frac c_ text volumetric rho . For pure homogeneous chemical compounds with established molecular or molar mass, or a molar quantity, heat capacity as an intensive property can be expressed on a per-mole basis instead of a per-mass basis by the following equations analogous to the per mass equations: C P , m = ( ∂ C ∂ n ) P displaystyle C_ P,m =left( frac partial C partial n right)_ P = molar heat capacity at constant pressure, C V , m = ( ∂ C ∂ n ) V displaystyle C_ V,m =left( frac partial C partial n right)_ V = molar heat capacity at constant volume, where n is the number of moles in the body or thermodynamic system.
One may refer to such a per-mole quantity as molar heat capacity to
distinguish it from specific heat capacity on a per-mass basis.
C i , m = ( ∂ C ∂ n ) displaystyle C_ i,m =left( frac partial C partial n right) = molar heat capacity at polytropic process. The most important polytropic processes run between the adiabatic and the isotherm functions, the polytropic index is between 1 and the adiabatic exponent (γ or κ). Dimensionless heat capacity[edit] The dimensionless heat capacity of a material is C ∗ = C n R = C N k , displaystyle C^ * = frac C nR = frac C Nk , where C is the heat capacity of a body made of the material in question
(J/K),
n is the amount of substance in the body (mol),
R is the gas constant (J/(K⋅mol)),
N is the number of molecules in the body (dimensionless),
k is
In the ideal gas article, dimensionless heat capacity C ∗ displaystyle C^ * is expressed as c ^ displaystyle hat c and is related there directly to half the number of degrees of freedom per particle. This holds true for quadratic degrees of freedom, a consequence of the equipartition theorem. More generally, the dimensionless heat capacity relates the logarithmic increase in temperature to the increase in the dimensionless entropy per particle S ∗ = S / N k displaystyle S^ * =S/Nk , measured in nats. C ∗ = d S ∗ d ( ln T ) . displaystyle C^ * = frac text d S^ * text d (ln T) . Alternatively, using base-2 logarithms, C* relates the base-2
logarithmic increase in temperature to the increase in the
dimensionless entropy measured in bits.[12]
T d S = δ Q , displaystyle T, text d S=delta Q, the absolute entropy can be calculated by integrating from zero to the final temperature Tf: S ( T f ) = ∫ T = 0 T f δ Q T = ∫ 0 T f δ Q d T d T T = ∫ 0 T f C ( T ) d T T . displaystyle S(T_ text f )=int _ T=0 ^ T_ text f frac delta Q T =int _ 0 ^ T_ text f frac delta Q text d T frac text d T T =int _ 0 ^ T_ text f C(T), frac text d T T . The heat capacity must be zero at zero temperature in order for the
above integral not to yield an infinite absolute entropy, which would
violate the third law of thermodynamics. One of the strengths of the
U pot = − 2 U kin . displaystyle U_ text pot =-2U_ text kin . The total energy U (= Upot + Ukin) therefore obeys U = − U kin . displaystyle U=-U_ text kin . If the system loses energy, for example, by radiating energy away into space, the average kinetic energy actually increases. If a temperature is defined by the average kinetic energy, then the system therefore can be said to have a negative heat capacity.[16] A more extreme version of this occurs with black holes. According to black-hole thermodynamics, the more mass and energy a black hole absorbs, the colder it becomes. In contrast, if it is a net emitter of energy, through Hawking radiation, it will become hotter and hotter until it boils away. Theory[edit] Factors that affect specific heat capacity[edit] Molecules undergo many characteristic internal vibrations. Potential energy stored in these internal degrees of freedom contributes to a sample’s energy content, [17] [18] but not to its temperature. More internal degrees of freedom tend to increase a substance's specific heat capacity, so long as temperatures are high enough to overcome quantum effects. For any given substance, the heat capacity of a body is directly proportional to the amount of substance it contains (measured in terms of mass or moles or volume). Doubling the amount of substance in a body doubles its heat capacity, etc. However, when this effect has been corrected for, by dividing the heat capacity by the quantity of substance in a body, the resulting specific heat capacity is a function of the structure of the substance itself. In particular, it depends on the number of degrees of freedom that are available to the particles in the substance; each independent degree of freedom allows the particles to store thermal energy. The translational kinetic energy of substance particles which manifests as temperature change is only one of the many possible degrees of freedom, and thus the larger the number of degrees of freedom available to the particles of a substance other than translational kinetic energy, the larger will be the specific heat capacity for the substance. For example, rotational kinetic energy of gas molecules stores heat energy in a way that increases heat capacity, since this energy does not contribute to temperature. In addition, quantum effects require that whenever energy be stored in any mechanism associated with a bound system which confers a degree of freedom, it must be stored in certain minimal-sized deposits (quanta) of energy, or else not stored at all. Such effects limit the full ability of some degrees of freedom to store energy when their lowest energy storage quantum amount is not easily supplied at the average energy of particles at a given temperature. In general, for this reason, specific heat capacities tend to fall at lower temperatures where the average thermal energy available to each particle degree of freedom is smaller, and thermal energy storage begins to be limited by these quantum effects. Due to this process, as temperature falls toward absolute zero, so also does heat capacity. Degrees of freedom[edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2015) (Learn how and when to remove this template message) Main article: degrees of freedom (physics and chemistry)
Molecules are quite different from the monatomic gases like helium and
argon. With monatomic gases, thermal energy comprises only
translational motions. Translational motions are ordinary, whole-body
movements in 3D space whereby particles move about and exchange energy
in collisions—like rubber balls in a vigorously shaken container
(see animation here [19]). These simple movements in the three
dimensions of space mean individual atoms have three translational
degrees of freedom. A degree of freedom is any form of energy in which
heat transferred into an object can be stored. This can be in
translational kinetic energy, rotational kinetic energy, or other
forms such as potential energy in vibrational modes. Only three
translational degrees of freedom (corresponding to the three
independent directions in space) are available for any individual
atom, whether it is free, as a monatomic molecule, or bound into a
polyatomic molecule.
As to rotation about an atom's axis (again, whether the atom is bound
or free), its energy of rotation is proportional to the moment of
inertia for the atom, which is extremely small compared to moments of
inertia of collections of atoms. This is because almost all of the
mass of a single atom is concentrated in its nucleus, which has a
radius too small to give a significant moment of inertia. In contrast,
the spacing of quantum energy levels for a rotating object is
inversely proportional to its moment of inertia, and so this spacing
becomes very large for objects with very small moments of inertia. For
these reasons, the contribution from rotation of atoms on their axes
is essentially zero in monatomic gases, because the energy spacing of
the associated quantum levels is too large for significant thermal
energy to be stored in rotation of systems with such small moments of
inertia. For similar reasons, axial rotation around bonds joining
atoms in diatomic gases (or along the linear axis in a linear molecule
of any length) can also be neglected as a possible "degree of freedom"
as well, since such rotation is similar to rotation of monatomic
atoms, and so occurs about an axis with a moment of inertia too small
to be able to store significant heat energy.
In polyatomic molecules, other rotational modes may become active, due
to the much higher moments of inertia about certain axes which do not
coincide with the linear axis of a linear molecule. These modes take
the place of some translational degrees of freedom for individual
atoms, since the atoms are moving in 3-D space, as the molecule
rotates. The narrowing of quantum mechanically determined energy
spacing between rotational states results from situations where atoms
are rotating around an axis that does not connect them, and thus form
an assembly that has a large moment of inertia. This small difference
between energy states allows the kinetic energy of this type of
rotational motion to store heat energy at ambient temperatures.
Furthermore, internal vibrational degrees of freedom also may become
active (these are also a type of translation, as seen from the view of
each atom). In summary, molecules are complex objects with a
population of atoms that may move about within the molecule in a
number of different ways (see animation at right), and each of these
ways of moving is capable of storing energy if the temperature is
sufficient.
The heat capacity of molecular substances (on a "per-atom" or
atom-molar, basis) does not exceed the heat capacity of monatomic
gases, unless vibrational modes are brought into play. The reason for
this is that vibrational modes allow energy to be stored as potential
energy in inter-atomic bonds in a molecule, which are not available to
atoms in monatomic gases. Up to about twice as much energy (on a
per-atom basis) per unit of temperature increase can be stored in a
solid as in a monatomic gas, by this mechanism of storing energy in
the potentials of interatomic bonds. This gives many solids about
twice the atom-molar heat capacity at room temperature of monatomic
gases.
However, quantum effects heavily affect the actual ratio at lower
temperatures (i.e., much lower than the melting temperature of the
solid), especially in solids with light and tightly bound atoms (e.g.,
beryllium metal or diamond). Polyatomic gases store intermediate
amounts of energy, giving them a "per-atom" heat capacity that is
between that of monatomic gases (3⁄2 R per mole of atoms, where R
is the ideal gas constant), and the maximum of fully excited warmer
solids (3 R per mole of atoms). For gases, heat capacity never falls
below the minimum of 3⁄2 R per mole (of molecules), since the
kinetic energy of gas molecules is always available to store at least
this much thermal energy. However, at cryogenic temperatures in
solids, heat capacity falls toward zero, as temperature approaches
absolute zero.
Example of temperature-dependent specific heat capacity, in a diatomic
gas[edit]
To illustrate the role of various degrees of freedom in storing heat,
we may consider nitrogen, a diatomic molecule that has five active
degrees of freedom at room temperature: the three comprising
translational motions plus two rotational degrees of freedom
internally. Although the constant-volume molar heat capacity of
nitrogen at this temperature is five-thirds that of monatomic gases,
on a per-mole of atoms basis, it is five-sixths that of a monatomic
gas. The reason for this is the loss of a degree of freedom due to the
bond when it does not allow storage of thermal energy. Two separate
nitrogen atoms would have a total of six degrees of freedom—the
three translational degrees of freedom of each atom. When the atoms
are bonded the molecule will still only have three translational
degrees of freedom, as the two atoms in the molecule move as one.
However, the molecule cannot be treated as a point object, and the
moment of inertia has increased sufficiently about two axes to allow
two rotational degrees of freedom to be active at room temperature to
give five degrees of freedom. The moment of inertia about the third
axis remains small, as this is the axis passing through the centres of
the two atoms, and so is similar to the small moment of inertia for
atoms of a monatomic gas. Thus, this degree of freedom does not act to
store heat, and does not contribute to the heat capacity of nitrogen.
The heat capacity per atom for nitrogen (5/2 R per mole molecules =
5/4 R per mole atoms) is therefore less than for a monatomic gas (3/2
R per mole molecules or atoms), so long as the temperature remains low
enough that no vibrational degrees of freedom are activated.[20]
At higher temperatures, however, nitrogen gas gains one more degree of
internal freedom, as the molecule is excited into higher vibrational
modes that store thermal energy. A vibrational degree of freedom
contributes a heat capacity of 1/2 R each for kinetic and potential
energy, for a total of R. Now the bond is contributing heat capacity,
and (because of storage of energy in potential energy) is contributing
more than if the atoms were not bonded. With full thermal excitation
of bond vibration, the heat capacity per volume, or per mole of gas
molecules approaches seven-thirds that of monatomic gases.
Significantly, this is seven-sixths of the monatomic gas value on a
mole-of-atoms basis, so this is now a higher heat capacity per atom
than the monatomic figure, because the vibrational mode enables for
diatomic gases allows an extra degree of potential energy freedom per
pair of atoms, which monatomic gases cannot possess.[21][22] See
thermodynamic temperature for more information on translational
motions, kinetic (heat) energy, and their relationship to temperature.
However, even at these large temperatures where gaseous nitrogen is
able to store 7/6ths of the energy per atom of a monatomic gas (making
it more efficient at storing energy on an atomic basis), it still only
stores 7/12 ths of the maximal per-atom heat capacity of a solid,
meaning it is not nearly as efficient at storing thermal energy on an
atomic basis, as solid substances can be. This is typical of gases,
and results because many of the potential bonds which might be storing
potential energy in gaseous nitrogen (as opposed to solid nitrogen)
are lacking, because only one of the spatial dimensions for each
nitrogen atom offers a bond into which potential energy can be stored
without increasing the kinetic energy of the atom. In general, solids
are most efficient, on an atomic basis, at storing thermal energy
(that is, they have the highest per-atom or per-mole-of-atoms heat
capacity).
Per mole of different units[edit]
Per mole of molecules[edit]
When the specific heat capacity, c, of a material is measured
(lowercase c means the unit quantity is in terms of mass), different
values arise because different substances have different molar masses
(essentially, the weight of the individual atoms or molecules). In
solids, thermal energy arises due to the number of atoms that are
vibrating. "Molar" heat capacity per mole of molecules, for both gases
and solids, offer figures which are arbitrarily large, since molecules
may be arbitrarily large. Such heat capacities are thus not intensive
quantities for this reason, since the quantity of mass being
considered can be increased without limit.
Per mole of atoms[edit]
Conversely, for molecular-based substances (which also absorb heat
into their internal degrees of freedom), massive, complex molecules
with high atomic count—like octane—can store a great deal of
energy per mole and yet are quite unremarkable on a mass basis, or on
a per-atom basis. This is because, in fully excited systems, heat is
stored independently by each atom in a substance, not primarily by the
bulk motion of molecules.
Thus, it is the heat capacity per-mole-of-atoms, not
per-mole-of-molecules, which is the intensive quantity, and which
comes closest to being a constant for all substances at high
temperatures. This relationship was noticed empirically in 1819, and
is called the Dulong–Petit law, after its two discoverers.[23]
Historically, the fact that specific heat capacities are approximately
equal when corrected by the presumed weight of the atoms of solids,
was an important piece of data in favor of the atomic theory of
matter.
Because of the connection of heat capacity to the number of atoms,
some care should be taken to specify a mole-of-molecules basis vs. a
mole-of-atoms basis, when comparing specific heat capacities of
molecular solids and gases. Ideal gases have the same numbers of
molecules per volume, so increasing molecular complexity adds heat
capacity on a per-volume and per-mole-of-molecules basis, but may
lower or raise heat capacity on a per-atom basis, depending on whether
the temperature is sufficient to store energy as atomic vibration.
In solids, the quantitative limit of heat capacity in general is about
3 R per mole of atoms, where R is the ideal gas constant. This 3 R
value is about 24.9 J/mole.K. Six degrees of freedom (three kinetic
and three potential) are available to each atom. Each of these six
contributes 1⁄2R specific heat capacity per mole of atoms.[24]
This limit of 3 R per mole specific heat capacity is approached at
room temperature for most solids, with significant departures at this
temperature only for solids composed of the lightest atoms which are
bound very strongly, such as beryllium (where the value is only of 66%
of 3 R), or diamond (where it is only 24% of 3 R). These large
departures are due to quantum effects which prevent full distribution
of heat into all vibrational modes, when the energy difference between
vibrational quantum states is very large compared to the average
energy available to each atom from the ambient temperature.
For monatomic gases, the specific heat is only half of 3 R per mole,
i.e. (3⁄2R per mole) due to loss of all potential energy degrees
of freedom in these gases. For polyatomic gases, the heat capacity
will be intermediate between these values on a per-mole-of-atoms
basis, and (for heat-stable molecules) would approach the limit of 3 R
per mole of atoms, for gases composed of complex molecules, and at
higher temperatures at which all vibrational modes accept excitational
energy. This is because very large and complex gas molecules may be
thought of as relatively large blocks of solid matter which have lost
only a relatively small fraction of degrees of freedom, as compared to
a fully integrated solid.
For a list of heat capacities per atom-mole of various substances, in
terms of R, see the last column of the table of heat capacities below.
Corollaries of these considerations for solids (volume-specific heat
capacity)[edit]
Since the bulk density of a solid chemical element is strongly
related to its molar mass (usually about 3 R per mole, as noted
above), there exists a noticeable inverse correlation between a
solid’s density and its specific heat capacity on a per-mass basis.
This is due to a very approximate tendency of atoms of most elements
to be about the same size (and constancy of mole-specific heat
capacity) resulting in a good correlation between the volume of any
given solid chemical element and its total heat capacity. Another way
of stating this, is that the volume-specific heat capacity (volumetric
heat capacity) of solid elements is roughly a constant. The molar
volume of solid elements is very roughly constant, and (even more
reliably) so also is the molar heat capacity for most solid
substances. These two factors determine the volumetric heat capacity,
which as a bulk property may be striking in consistency. For example,
the element uranium is a metal which has a density almost 36 times
that of the metal lithium, but uranium's specific heat capacity on a
volumetric basis (i.e. per given volume of metal) is only 18% larger
than lithium's.
Since the volume-specific corollary of the Dulong–Petit specific
heat capacity relationship requires that atoms of all elements take up
(on average) the same volume in solids, there are many departures from
it, with most of these due to variations in atomic size. For instance,
arsenic, which is only 14.5% less dense than antimony, has nearly 59%
more specific heat capacity on a mass basis. In other words; even
though an ingot of arsenic is only about 17% larger than an antimony
one of the same mass, it absorbs about 59% more heat for a given
temperature rise. The heat capacity ratios of the two substances
closely follows the ratios of their molar volumes (the ratios of
numbers of atoms in the same volume of each substance); the departure
from the correlation to simple volumes in this case is due to lighter
arsenic atoms being significantly more closely packed than antimony
atoms, instead of similar size. In other words, similar-sized atoms
would cause a mole of arsenic to be 63% larger than a mole of
antimony, with a correspondingly lower density, allowing its volume to
more closely mirror its heat capacity behavior.
Other factors[edit]
C V = ( ∂ U ∂ T ) V = 3 2 N k B = 3 2 n R , displaystyle C_ V =left( frac partial U partial T right)_ V = frac 3 2 N,k_ B = frac 3 2 n,R, C V , m = C V n = 3 2 R , displaystyle C_ V,m = frac C_ V n = frac 3 2 R, where C V displaystyle C_ V is the heat capacity at constant volume of the gas, C V , m displaystyle C_ V,m is the molar heat capacity at constant volume of the gas, N is the total number of atoms present in the container, n is the number of moles of atoms present in the container (n is the ratio of N and Avogadro’s number), R is the ideal gas constant (8.3144621(75) J/(mol⋅K), equal to the product of Boltzmann’s constant k B displaystyle k_ text B and Avogadro’s number. The following table shows experimental molar constant-volume heat-capacity measurements taken for each noble monatomic gas (at 1 atm and 25 °C):
He 12.5 1.50 Ne 12.5 1.50 Ar 12.5 1.50 Kr 12.5 1.50 Xe 12.5 1.50 It is apparent from the table that the experimental heat capacities of the monatomic noble gases agrees with this simple application of statistical mechanics to a very high degree. The molar heat capacity of a monatomic gas at constant pressure is then C p , m = C V , m + R = 5 2 R . displaystyle C_ p,m =C_ V,m +R= frac 5 2 R.
Constant-volume specific heat capacity of a diatomic gas (idealised). As temperature increases, heat capacity goes from 3/2 R (translation contribution only), to 5/2 R (translation plus rotation), finally to a maximum of 7/2 R (translation + rotation + vibration) In the somewhat more complex case of an ideal gas of diatomic molecules, the presence of internal degrees of freedom are apparent. In addition to the three translational degrees of freedom, there are rotational and vibrational degrees of freedom. In general, the number of degrees of freedom, f, in a molecule with na atoms is 3na: f = 3 n a . displaystyle f=3n_ text a . Mathematically, there are a total of three rotational degrees of freedom, one corresponding to rotation about each of the axes of three-dimensional space. However, in practice only the existence of two degrees of rotational freedom for linear molecules will be considered. This approximation is valid because the moment of inertia about the internuclear axis is vanishingly small with respect to other moments of inertia in the molecule (this is due to the very small rotational moments of single atoms, due to the concentration of almost all their mass at their centers; compare also the extremely small radii of the atomic nuclei compared to the distance between them in a diatomic molecule). Quantum mechanically, it can be shown that the interval between successive rotational energy eigenstates is inversely proportional to the moment of inertia about that axis. Because the moment of inertia about the internuclear axis is vanishingly small relative to the other two rotational axes, the energy spacing can be considered so high that no excitations of the rotational state can occur unless the temperature is extremely high. It is easy to calculate the expected number of vibrational degrees of freedom (or vibrational modes). There are three degrees of translational freedom and two degrees of rotational freedom, therefore f vib = f − f trans − f rot = 6 − 3 − 2 = 1. displaystyle f_ text vib =f-f_ text trans -f_ text rot =6-3-2=1. Each rotational and translational degree of freedom will contribute R/2 in the total molar heat capacity of the gas. Each vibrational mode will contribute R displaystyle R to the total molar heat capacity, however. This is because for each vibrational mode, there is a potential and kinetic energy component. Both the potential and kinetic components will contribute R/2 to the total molar heat capacity of the gas. Therefore, a diatomic molecule would be expected to have a molar constant-volume heat capacity of C V , m = 3 R 2 + R + R = 7 R 2 = 3.5 R , displaystyle C_ V,m = frac 3R 2 +R+R= frac 7R 2 =3.5R, where the terms originate from the translational, rotational, and vibrational degrees of freedom respectively. Constant-volume specific heat capacity of diatomic gases (real gases)
between about 200 K and 2000 K. This temperature range is
not large enough to include both quantum transitions in all gases.
Instead, at 200 K, all but hydrogen are fully rotationally
excited, so all have at least 5/2 R heat capacity. (
The following is a table of some molar constant-volume heat capacities of various diatomic gases at standard temperature (25 °C = 298 K)
H2 20.18 2.427 CO 20.2 2.43 N2 19.9 2.39 Cl2 24.1 3.06 Br2 (vapour) 28.2 3.39 From the above table, clearly there is a problem with the above theory. All of the diatomics examined have heat capacities that are lower than those predicted by the equipartition theorem, except Br2. However, as the atoms composing the molecules become heavier, the heat capacities move closer to their expected values. One of the reasons for this phenomenon is the quantization of vibrational, and to a lesser extent, rotational states. In fact, if it is assumed that the molecules remain in their lowest-energy vibrational state because the inter-level energy spacings for vibration energies are large, the predicted molar constant-volume heat capacity for a diatomic molecule becomes just that from the contributions of translation and rotation: C V , m = 3 R 2 + R = 5 R 2 = 2.5 R , displaystyle C_ V,m = frac 3R 2 +R= frac 5R 2 =2.5R, which is a fairly close approximation of the heat capacities of the lighter molecules in the above table. If the quantum harmonic oscillator approximation is made, it turns out that the quantum vibrational energy level spacings are actually inversely proportional to the square root of the reduced mass of the atoms composing the diatomic molecule. Therefore, in the case of the heavier diatomic molecules such as chlorine or bromine, the quantum vibrational energy-level spacings become finer, which allows more excitations into higher vibrational levels at lower temperatures. This limit for storing heat capacity in vibrational modes, as discussed above, becomes 7R/2 = 3.5 R per mole of gas molecules, which is fairly consistent with the measured value for Br2 at room temperature. As temperatures rise, all diatomic gases approach this value. General gas phase[edit] This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2015) (Learn how and when to remove this template message) The specific heat of the gas is best conceptualized in terms of the degrees of freedom of an individual molecule. The different degrees of freedom correspond to the different ways in which the molecule may store energy. The molecule may store energy in its translational motion according to the formula: E = 1 2 m ( v x 2 + v y 2 + v z 2 ) displaystyle E= frac 1 2 ,mleft(v_ x ^ 2 +v_ y ^ 2 +v_ z ^ 2 right) where m is the mass of the molecule and [ v x , v y , v z ] displaystyle [v_ x ,v_ y ,v_ z ] is velocity of the center of mass of the molecule. Each direction of motion constitutes a degree of freedom, thus there are three translational degrees of freedom. In addition, a molecule may have rotational motion. The kinetic energy of rotational motion is generally expressed as E = 1 2 ( I 1 ω 1 2 + I 2 ω 2 2 + I 3 ω 3 2 ) displaystyle E= frac 1 2 ,left(I_ 1 omega _ 1 ^ 2 +I_ 2 omega _ 2 ^ 2 +I_ 3 omega _ 3 ^ 2 right) where I is the moment of inertia tensor of the molecule, and [ ω 1 , ω 2 , ω 3 ] displaystyle [omega _ 1 ,omega _ 2 ,omega _ 3 ] is the angular velocity pseudo-vector (in a coordinate system aligned with the principal axes of the molecule). In general, then, there will be three additional degrees of freedom corresponding to the rotational motion of the molecule, (For linear molecules one of the inertia tensor terms vanishes and there are only two rotational degrees of freedom). The degrees of freedom corresponding to translations and rotations are called the rigid degrees of freedom, since they do not involve any deformation of the molecule. The motions of the atoms in a molecule which are not part of its gross translational motion or rotation may be classified as vibrational motions. It can be shown that if there are n atoms in the molecule, there will be as many as v = 3 n − 3 − n r displaystyle v=3n-3-n_ r vibrational degrees of freedom, where n r displaystyle n_ r is the number of rotational degrees of freedom. A vibrational degree of freedom corresponds to a specific way in which all the atoms of a molecule can vibrate. The actual number of possible vibrations may be less than this maximal one, due to various symmetries. For example, triatomic nitrous oxide N2O will have only 2 degrees of rotational freedom (since it is a linear molecule) and contains n=3 atoms: thus the number of possible vibrational degrees of freedom will be v = (3⋅3) − 3 − 2 = 4. There are four ways or "modes" in which the three atoms can vibrate, corresponding to 1) A mode in which an atom at each end of the molecule moves away from, or towards, the center atom at the same time, 2) a mode in which either end atom moves asynchronously with regard to the other two, and 3) and 4) two modes in which the molecule bends out of line, from the center, in the two possible planar directions that are orthogonal to its axis. Each vibrational degree of freedom confers TWO total degrees of freedom, since vibrational energy mode partitions into 1 kinetic and 1 potential mode. This would give nitrous oxide 3 translational, 2 rotational, and 4 vibrational modes (but these last giving 8 vibrational degrees of freedom), for storing energy. This is a total of f = 3 + 2 + 8 = 13 total energy-storing degrees of freedom, for N2O. For a bent molecule like water H2O, a similar calculation gives 9 − 3 − 3 = 3 modes of vibration, and 3 (translational) + 3 (rotational) + 6 (vibrational) = 12 degrees of freedom. The storage of energy into degrees of freedom[edit] If the molecule could be entirely described using classical mechanics, then the theorem of equipartition of energy could be used to predict that each degree of freedom would have an average energy in the amount of (1/2)kT, where k is Boltzmann’s constant, and T is the temperature. Our calculation of the constant-volume heat capacity would be straightforward. Each molecule would be holding, on average, an energy of (f/2)kT, where f is the total number of degrees of freedom in the molecule. Note that Nk = R if N is Avogadro's number, which is the case in considering the heat capacity of a mole of molecules. Thus, the total internal energy of the gas would be (f/2)NkT, where N is the total number of molecules. The heat capacity (at constant volume) would then be a constant (f/2)Nk, the mole-specific heat capacity would be (f/2)R, the molecule-specific heat capacity would be (f/2)k, and the dimensionless heat capacity would be just f/2. Here again, each vibrational degree of freedom contributes 2f. Thus, a mole of nitrous oxide would have a total constant-volume heat capacity (including vibration) of (13/2)R by this calculation. In summary, the molar heat capacity (mole-specific heat capacity) of an ideal gas with f degrees of freedom is given by C V , m = f 2 R . displaystyle C_ V,m = frac f 2 R. This equation applies to all polyatomic gases, if the degrees of
freedom are known.[26]
The constant-pressure heat capacity for any gas would exceed this by
an extra R (see Mayer's relation, above). As example Cp would be a
total of (15/2)R for nitrous oxide.
The effect of quantum energy levels in storing energy in degrees of
freedom[edit]
The various degrees of freedom cannot generally be considered to obey
classical mechanics, however. Classically, the energy residing in each
degree of freedom is assumed to be continuous—it can take on any
positive value, depending on the temperature. In reality, the amount
of energy that may reside in a particular degree of freedom is
quantized: It may only be increased and decreased in finite amounts. A
good estimate of the size of this minimum amount is the energy of the
first excited state of that degree of freedom above its ground state.
For example, the first vibrational state of the hydrogen chloride
(HCl) molecule has an energy of about 5.74 × 10−20 joule.
If this amount of energy were deposited in a classical degree of
freedom, it would correspond to a temperature of about 4156 K.
If the temperature of the substance is so low that the equipartition
energy of (1/2)kT is much smaller than this excitation energy,
then there will be little or no energy in this degree of freedom. This
degree of freedom is then said to be “frozen out". As mentioned
above, the temperature corresponding to the first excited vibrational
state of HCl is about 4156 K. For temperatures well below this
value, the vibrational degrees of freedom of the HCl molecule will be
frozen out. They will contain little energy and will not contribute to
the thermal energy or the heat capacity of HCl gas.
This section does not cite any sources. Please help improve this section by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2015) (Learn how and when to remove this template message) It can be seen that for each degree of freedom there is a critical temperature at which the degree of freedom “unfreezes” and begins to accept energy in a classical way. In the case of translational degrees of freedom, this temperature is that temperature at which the thermal wavelength of the molecules is roughly equal to the size of the container. For a container of macroscopic size (e.g. 10 cm) this temperature is extremely small and has no significance, since the gas will certainly liquify or freeze before this low temperature is reached. For any real gas translational degrees of freedom may be considered to always be classical and contain an average energy of (3/2)kT per molecule. The rotational degrees of freedom are the next to “unfreeze". In a diatomic gas, for example, the critical temperature for this transition is usually a few tens of kelvins, although with a very light molecule such as hydrogen the rotational energy levels will be spaced so widely that rotational heat capacity may not completely "unfreeze" until considerably higher temperatures are reached. Finally, the vibrational degrees of freedom are generally the last to unfreeze. As an example, for diatomic gases, the critical temperature for the vibrational motion is usually a few thousands of kelvins, and thus for the nitrogen in our example at room temperature, no vibration modes would be excited, and the constant-volume heat capacity at room temperature is (5/2)R/mole, not (7/2)R/mole. As seen above, with some unusually heavy gases such as iodine gas I2, or bromine gas Br2, some vibrational heat capacity may be observed even at room temperatures. It should be noted that it has been assumed that atoms have no rotational or internal degrees of freedom. This is in fact untrue. For example, atomic electrons can exist in excited states and even the atomic nucleus can have excited states as well. Each of these internal degrees of freedom are assumed to be frozen out due to their relatively high excitation energy. Nevertheless, for sufficiently high temperatures, these degrees of freedom cannot be ignored. In a few exceptional cases, such molecular electronic transitions are of sufficiently low energy that they contribute to heat capacity at room temperature, or even at cryogenic temperatures. One example of an electronic transition degree of freedom which contributes heat capacity at standard temperature is that of nitric oxide (NO), in which the single electron in an anti-bonding molecular orbital has energy transitions which contribute to the heat capacity of the gas even at room temperature. An example of a nuclear magnetic transition degree of freedom which is of importance to heat capacity, is the transition which converts the spin isomers of hydrogen gas (H2) into each other. At room temperature, the proton spins of hydrogen gas are aligned 75% of the time, resulting in orthohydrogen when they are. Thus, some thermal energy has been stored in the degree of freedom available when parahydrogen (in which spins are anti-aligned) absorbs energy, and is converted to the higher energy ortho form. However, at the temperature of liquid hydrogen, not enough heat energy is available to produce orthohydrogen (that is, the transition energy between forms is large enough to "freeze out" at this low temperature), and thus the parahydrogen form predominates. The heat capacity of the transition is sufficient to release enough heat, as orthohydrogen converts to the lower-energy parahydrogen, to boil the hydrogen liquid to gas again, if this evolved heat is not removed with a catalyst after the gas has been cooled and condensed. This example also illustrates the fact that some modes of storage of heat may not be in constant equilibrium with each other in substances, and heat absorbed or released from such phase changes may "catch up" with temperature changes of substances, only after a certain time. In other words, the heat evolved and absorbed from the ortho-para isomeric transition contributes to the heat capacity of hydrogen on long time-scales, but not on short time-scales. These time scales may also depend on the presence of a catalyst. Less exotic phase-changes may contribute to the heat-capacity of substances and systems, as well, as (for example) when water is converted back and forth from solid to liquid or gas form. Phase changes store heat energy entirely in breaking the bonds of the potential energy interactions between molecules of a substance. As in the case of hydrogen, it is also possible for phase changes to be hindered as the temperature drops, thus they do not catch up and become apparent, without a catalyst. For example, it is possible to supercool liquid water to below the freezing point, and not observe the heat evolved when the water changes to ice, so long as the water remains liquid. This heat appears instantly when the water freezes. Solid phase[edit] Main articles: Einstein solid, Debye model, and Kinetic theory of solids The dimensionless heat capacity divided by three, as a function of
temperature as predicted by the
For matter in a crystalline solid phase, the Dulong–Petit law, which
was discovered empirically, states that the molar heat capacity
assumes the value 3 R. Indeed, for solid metallic chemical elements at
room temperature, molar heat capacities range from about 2.8 R to 3.4
R. Large exceptions at the lower end involve solids composed of
relatively low-mass, tightly bonded atoms, such as beryllium at 2.0 R,
and diamond at only 0.735 R. The latter conditions create larger
quantum vibrational energy spacing, thus many vibrational modes have
energies too high to be populated (and thus are "frozen out") at room
temperature. At the higher end of possible heat capacities, heat
capacity may exceed R by modest amounts, due to contributions from
anharmonic vibrations in solids, and sometimes a modest contribution
from conduction electrons in metals. These are not degrees of freedom
treated in the Einstein or Debye theories.
The theoretical maximum heat capacity for multi-atomic gases at higher
temperatures, as the molecules become larger, also approaches the
Dulong–Petit limit of 3 R, so long as this is calculated per mole of
atoms, not molecules. The reason for this behavior is that, in theory,
gases with very large molecules have almost the same high-temperature
heat capacity as solids, lacking only the (small) heat capacity
contribution that comes from potential energy that cannot be stored
between separate molecules in a gas.
The Dulong–Petit limit results from the equipartition theorem, and
as such is only valid in the classical limit of a microstate
continuum, which is a high temperature limit. For light and
non-metallic elements, as well as most of the common molecular solids
based on carbon compounds at standard ambient temperature, quantum
effects may also play an important role, as they do in multi-atomic
gases. These effects usually combine to give heat capacities lower
than 3 R per mole of atoms in the solid, although in molecular solids,
heat capacities calculated per mole of molecules in molecular solids
may be more than 3 R. For example, the heat capacity of water ice at
the melting point is about 4.6 R per mole of molecules, but only 1.5 R
per mole of atoms. As noted, heat capacity values far lower than 3 R
"per atom" (as is the case with diamond and beryllium) result from
“freezing out” of possible vibration modes for light atoms at
suitably low temperatures, just as happens in many low-mass-atom gases
at room temperatures (where vibrational modes are all frozen out).
Because of high crystal binding energies, the effects of vibrational
mode freezing are observed in solids more often than liquids: for
example the heat capacity of liquid water is twice that of ice at near
the same temperature, and is again close to the 3 R per mole of atoms
of the Dulong–Petit theoretical maximum.
This section needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed. (February 2015) (Learn how and when to remove this template message) See also: List of thermal conductivities
Note that the especially high molar values, as for paraffin, gasoline,
water and ammonia, result from calculating specific heats in terms of
moles of molecules. If specific heat is expressed per mole of atoms
for these substances, none of the constant-volume values exceed, to
any large extent, the theoretical Dulong–Petit limit of 25
J⋅mol−1⋅K−1 = 3 R per mole of atoms (see the last column of
this table). Paraffin, for example, has very large molecules and thus
a high heat capacity per mole, but as a substance it does not have
remarkable heat capacity in terms of volume, mass, or atom-mol (which
is just 1.41 R per mole of atoms, or less than half of most solids, in
terms of heat capacity per atom).
In the last column, major departures of solids at standard
temperatures from the
Table of specific heat capacities at 25 °C (298 K) unless otherwise noted.[citation needed] Notable minima and maxima are shown in maroon Substance Phase Isobaric mass heat capacity cP J⋅g−1⋅K−1 Isobaric molar heat capacity CP,m J⋅mol−1⋅K−1 Isochore molar heat capacity CV,m J⋅mol−1⋅K−1 Isobaric volumetric heat capacity CP,v J⋅cm−3⋅K−1 Isochore atom-molar heat capacity in units of R CV,am atom-mol−1 Air (Sea level, dry, 0 °C (273.15 K)) gas 1.0035 29.07 20.7643 0.001297 ~ 1.25 R Air (typical room conditionsA) gas 1.012 29.19 20.85 0.00121 ~ 1.25 R Aluminium solid 0.897 24.2 2.422 2.91 R Ammonia liquid 4.700 80.08 3.263 3.21 R Animal tissue (incl. human)[30] mixed 3.5 3.7* Antimony solid 0.207 25.2 1.386 3.03 R Argon gas 0.5203 20.7862 12.4717 1.50 R Arsenic solid 0.328 24.6 1.878 2.96 R Beryllium solid 1.82 16.4 3.367 1.97 R Bismuth[31] solid 0.123 25.7 1.20 3.09 R Cadmium solid 0.231 26.02 3.13 R
1.14 R Chromium solid 0.449 23.35 2.81 R Copper solid 0.385 24.47 3.45 2.94 R Diamond solid 0.5091 6.115 1.782 0.74 R Ethanol liquid 2.44 112 1.925 1.50 R
1.64 1.05 R Glass[31] solid 0.84 2.1 Gold solid 0.129 25.42 2.492 3.05 R Granite[31] solid 0.790 2.17 Graphite solid 0.710 8.53 1.534 1.03 R Helium gas 5.1932 20.7862 12.4717 1.50 R Hydrogen gas 14.30 28.82 1.23 R
1.05 R Iron solid 0.412 25.09[32] 3.537 3.02 R Lead solid 0.129 26.4 1.44 3.18 R Lithium solid 3.58 24.8 1.912 2.98 R
2.242 3.65 R Magnesium solid 1.02 24.9 1.773 2.99 R Mercury liquid 0.1395 27.98 1.888 3.36 R
0.85 R Methanol[34] liquid 2.14 68.62 1.38 R
2.62 Nitrogen gas 1.040 29.12 20.8 1.25 R Neon gas 1.0301 20.7862 12.4717 1.50 R Oxygen gas 0.918 29.38 21.0 1.26 R Paraffin wax C25H52 solid 2.5 (ave) 900 2.325 1.41 R Polyethylene (rotomolding grade)[36][37] solid 2.3027 Silica (fused) solid 0.703 42.2 1.547 1.69 R Silver[31] solid 0.233 24.9 2.44 2.99 R Sodium solid 1.230 28.23 3.39 R Steel solid 0.466 3.756 Tin solid 0.227 27.112 1.659 3.26 R Titanium solid 0.523 26.060 2.6384 3.13 R Tungsten[31] solid 0.134 24.8 2.58 2.98 R Uranium solid 0.116 27.7 2.216 3.33 R
1.12 R
1.938 1.53 R Zinc[31] solid 0.387 25.2 2.76 3.03 R Substance Phase Isobaric mass heat capacity cP J⋅g−1⋅K−1 Isobaric molar heat capacity CP,m J⋅mol−1⋅K−1 Isochore molar heat capacity CV,m J⋅mol−1⋅K−1 Isobaric volumetric heat capacity CP,v J⋅cm−3⋅K−1 Isochore atom-molar heat capacity in units of R CV,am atom-mol−1 A Assuming an altitude of 194 metres above mean sea level (the
world–wide median altitude of human habitation), an indoor
temperature of 23 °C, a dewpoint of 9 °C (40.85% relative
humidity), and 760 mm–Hg sea level–corrected barometric
pressure (molar water vapor content = 1.16%).
*Derived data by calculation. This is for water-rich tissues such as
brain. The whole-body average figure for mammals is approximately 2.9
J⋅cm−3⋅K−1 [38]
Substance Phase cP J⋅g−1⋅K−1 Asphalt solid 0.920 Brick solid 0.840 Concrete solid 0.880 Glass, silica solid 0.840 Glass, crown solid 0.670 Glass, flint solid 0.503 Glass, pyrex solid 0.753 Granite solid 0.790 Gypsum solid 1.090 Marble, mica solid 0.880 Sand solid 0.835 Soil solid 0.800 Water liquid 4.1813 Wood solid 1.7 (1.2 to 2.9) Substance Phase cP J g−1 K−1 See also[edit] Physics portal Quantum statistical mechanics
Notes[edit] ^ a b IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "Standard Pressure".. Besides being a round number, this had a very practical effect: relatively few[quantify] people live and work at precisely sea level; 100 kPa equates to the mean pressure at an altitude of about 112 metres (which is closer to the 194–metre, world–wide median altitude of human habitation[citation needed]). References[edit] ^ Halliday, David; Resnick, Robert (2013). Fundamentals of Physics.
Wiley. p. 524.
^ Kittel, Charles (2005). Introduction to Solid State Physics (8th
ed.). Hoboken, New Jersey, USA: John Wiley & Sons. p. 141.
ISBN 0-471-41526-X.
^ Blundell, Stephen (2001). Magnetism in Condensed Matter. Oxford
Master Series in Condensed Matter Physics (1st ed.). Hoboken, New
Jersey, USA: Oxford University Press. p. 27.
ISBN 978-0-19-850591-4.
^ Kittel, Charles (2005). Introduction to Solid State Physics (8th
ed.). Hoboken, New Jersey, USA: John Wiley & Sons. p. 141.
ISBN 0-471-41526-X.
^ Laider, Keith J. (1993). The World of Physical Chemistry. Oxford
University Press. ISBN 0-19-855919-4.
^ International Union of Pure and Applied Chemistry, Physical
Chemistry Division. "Quantities, Units and Symbols in Physical
Chemistry" (PDF). Blackwell Sciences. p. 7. The adjective
specific before the name of an extensive quantity is often used to
mean divided by mass.
^
Further reading[edit] Encyclopædia Britannica, 2015, "
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