An ideal gas is a theoretical
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
composed of many randomly moving
point particle
A point particle (ideal particle or point-like particle, often spelled pointlike particle) is an idealization of particles heavily used in physics. Its defining feature is that it lacks spatial extension; being dimensionless, it does not take up ...
s that are not subject to
interparticle interactions.
The ideal gas concept is useful because it obeys the
ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
, a simplified
equation of state
In physics, chemistry, and thermodynamics, an equation of state is a thermodynamic equation relating state variables, which describe the state of matter under a given set of physical conditions, such as pressure, volume, temperature, or internal ...
, and is amenable to analysis under
statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
. The requirement of zero interaction can often be relaxed if, for example, the interaction is
perfectly elastic or regarded as point-like collisions.
Under various conditions of temperature and pressure, many
real gas
Real gases are nonideal gases whose molecules occupy space and have interactions; consequently, they do not adhere to the ideal gas law.
To understand the behaviour of real gases, the following must be taken into account:
*compressibility effect ...
es behave qualitatively like an ideal gas where the gas molecules (or atoms for
monatomic gas
In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
) play the role of the ideal particles. Many gases such as
nitrogen
Nitrogen is the chemical element with the symbol N and atomic number 7. Nitrogen is a nonmetal and the lightest member of group 15 of the periodic table, often called the pnictogens. It is a common element in the universe, estimated at se ...
,
oxygen
Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements as wel ...
,
hydrogen
Hydrogen is the chemical element with the symbol H and atomic number 1. Hydrogen is the lightest element. At standard conditions hydrogen is a gas of diatomic molecules having the formula . It is colorless, odorless, tasteless, non-toxic, an ...
,
noble gas
The noble gases (historically also the inert gases; sometimes referred to as aerogens) make up a class of chemical elements with similar properties; under standard conditions, they are all odorless, colorless, monatomic gases with very low chemi ...
es, some heavier gases like
carbon dioxide
Carbon dioxide (chemical formula ) is a chemical compound made up of molecules that each have one carbon atom covalently double bonded to two oxygen atoms. It is found in the gas state at room temperature. In the air, carbon dioxide is transpar ...
and mixtures such as air, can be treated as ideal gases within reasonable tolerances
over a considerable parameter range around
standard temperature and pressure
Standard temperature and pressure (STP) are standard sets of conditions for experimental measurements to be established to allow comparisons to be made between different sets of data. The most used standards are those of the International Union o ...
. Generally, a gas behaves more like an ideal gas at higher
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
and lower
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
,
[ as the ]potential energy
In physics, potential energy is the energy held by an object because of its position relative to other objects, stresses within itself, its electric charge, or other factors.
Common types of potential energy include the gravitational potentia ...
due to intermolecular forces becomes less significant compared with the particles' kinetic energy
In physics, the kinetic energy of an object is the energy that it possesses due to its motion.
It is defined as the work needed to accelerate a body of a given mass from rest to its stated velocity. Having gained this energy during its accele ...
, and the size of the molecules becomes less significant compared to the empty space between them. One mole
Mole (or Molé) may refer to:
Animals
* Mole (animal) or "true mole", mammals in the family Talpidae, found in Eurasia and North America
* Golden moles, southern African mammals in the family Chrysochloridae, similar to but unrelated to Talpida ...
of an ideal gas has a volume of 22.710947(13) litres at standard temperature and pressure (a temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
of 273.15 K and an absolute pressure
Pressure measurement is the measurement of an applied force by a fluid (liquid or gas) on a surface. Pressure is typically measured in units of force per unit of surface area. Many techniques have been developed for the measurement of pressur ...
of exactly 105 Pa) as defined by IUPAC
The International Union of Pure and Applied Chemistry (IUPAC ) is an international federation of National Adhering Organizations working for the advancement of the chemical sciences, especially by developing nomenclature and terminology. It is ...
since 1982.
The ideal gas model tends to fail at lower temperatures or higher pressures, when intermolecular forces and molecular size becomes important. It also fails for most heavy gases, such as many refrigerant
A refrigerant is a working fluid used in the heat pump and refrigeration cycle, refrigeration cycle of air conditioning systems and heat pumps where in most cases they undergo a repeated phase transition from a liquid to a gas and back again. Ref ...
s,[ and for gases with strong intermolecular forces, notably ]water vapor
(99.9839 °C)
, -
, Boiling point
,
, -
, specific gas constant
, 461.5 J/( kg·K)
, -
, Heat of vaporization
, 2.27 MJ/kg
, -
, Heat capacity
, 1.864 kJ/(kg·K)
Water vapor, water vapour or aqueous vapor is the gaseous pha ...
. At high pressures, the volume of a real gas is often considerably larger than that of an ideal gas. At low temperatures, the pressure of a real gas is often considerably less than that of an ideal gas. At some point of low temperature and high pressure, real gases undergo a phase transition
In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
, such as to a liquid
A liquid is a nearly incompressible fluid that conforms to the shape of its container but retains a (nearly) constant volume independent of pressure. As such, it is one of the four fundamental states of matter (the others being solid, gas, a ...
or a solid
Solid is one of the State of matter#Four fundamental states, four fundamental states of matter (the others being liquid, gas, and Plasma (physics), plasma). The molecules in a solid are closely packed together and contain the least amount o ...
. The model of an ideal gas, however, does not describe or allow phase transitions. These must be modeled by more complex equations of state. The deviation from the ideal gas behavior can be described by a dimensionless quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, the compressibility factor
In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to ...
, .
The ideal gas model has been explored in both the Newtonian dynamics
In physics, Newtonian dynamics (also known as Newtonian mechanics) is the study of the dynamics of a particle or a small body according to Newton's laws of motion.
Mathematical generalizations
Typically, the Newtonian dynamics occurs in a thre ...
(as in "kinetic theory
Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to:
* Kinetic theory, describing a gas as particles in random motion
* Kinetic energy, the energy of an object that it possesses due to its motion
Art and ente ...
") and in quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
(as a "gas in a box
In quantum mechanics, the results of the quantum particle in a box can be used to look at the equilibrium situation for a quantum ideal gas in a box which is a box containing a large number of molecules which do not interact with each other exce ...
"). The ideal gas model has also been used to model the behavior of electrons in a metal (in the Drude model and the free electron model
In solid-state physics, the free electron model is a quantum mechanical model for the behaviour of charge carriers in a metallic solid. It was developed in 1927, principally by Arnold Sommerfeld, who combined the classical Drude model with quant ...
), and it is one of the most important models in statistical mechanics.
If the pressure of an ideal gas is reduced in a throttling process the temperature of the gas does not change. (If the pressure of a real gas is reduced in a throttling process, its temperature either falls or rises, depending on whether its Joule–Thomson coefficient is positive or negative.)
Types of ideal gas
There are three basic classes of ideal gas:
* the classical or Maxwell–Boltzmann ideal gas,
* the ideal quantum Bose gas
An ideal Bose gas is a quantum-mechanical phase of matter, analogous to a classical ideal gas. It is composed of bosons, which have an integer value of spin, and abide by Bose–Einstein statistics. The statistical mechanics of bosons were deve ...
, composed of boson
In particle physics, a boson ( ) is a subatomic particle whose spin quantum number has an integer value (0,1,2 ...). Bosons form one of the two fundamental classes of subatomic particle, the other being fermions, which have odd half-integer s ...
s, and
* the ideal quantum Fermi gas
An ideal Fermi gas is a state of matter which is an ensemble of many non-interacting fermions. Fermions are particles that obey Fermi–Dirac statistics, like electrons, protons, and neutrons, and, in general, particles with half-integer spin. ...
, composed of fermion
In particle physics, a fermion is a particle that follows Fermi–Dirac statistics. Generally, it has a half-odd-integer spin: spin , spin , etc. In addition, these particles obey the Pauli exclusion principle. Fermions include all quarks an ...
s.
The classical ideal gas can be separated into two types: The classical thermodynamic ideal gas and the ideal quantum Boltzmann gas. Both are essentially the same, except that the classical thermodynamic ideal gas is based on classical statistical mechanics
In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...
, and certain thermodynamic parameters such as the entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
are only specified to within an undetermined additive constant. The ideal quantum Boltzmann gas overcomes this limitation by taking the limit of the quantum Bose gas and quantum Fermi gas in the limit of high temperature to specify these additive constants. The behavior of a quantum Boltzmann gas is the same as that of a classical ideal gas except for the specification of these constants. The results of the quantum Boltzmann gas are used in a number of cases including the Sackur–Tetrode equation The Sackur–Tetrode equation is an expression for the entropy of a monatomic ideal gas.
It is named for Hugo Martin Tetrode (1895–1931) and Otto Sackur (1880–1914), who developed it independently as a solution of Boltzmann's gas statistics and ...
for the entropy of an ideal gas and the Saha ionization equation
In physics, the Saha ionization equation is an expression that relates the ionization state of a gas in thermal equilibrium to the temperature and pressure. The equation is a result of combining ideas of quantum mechanics and statistical mechanics ...
for a weakly ionized plasma
Plasma or plasm may refer to:
Science
* Plasma (physics), one of the four fundamental states of matter
* Plasma (mineral), a green translucent silica mineral
* Quark–gluon plasma, a state of matter in quantum chromodynamics
Biology
* Blood pla ...
.
Classical thermodynamic ideal gas
The classical thermodynamic properties of an ideal gas can be described by two equations of state:
Ideal gas law
The ideal gas law
The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stat ...
is the equation of state for an ideal gas, given by:
where
* is the pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country and e ...
* is the volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). The de ...
* is the amount of substance
In chemistry, the amount of substance ''n'' in a given sample of matter is defined as the quantity or number of discrete atomic-scale particles in it divided by the Avogadro constant ''N''A. The particles or entities may be molecules, atoms, ions, ...
of the gas (in moles Moles can refer to:
* Moles de Xert, a mountain range in the Baix Maestrat comarca, Valencian Community, Spain
* The Moles (Australian band)
*The Moles, alter ego of Scottish band Simon Dupree and the Big Sound
People
*Abraham Moles, French engin ...
)
* is the absolute temperature
Thermodynamic temperature is a quantity defined in thermodynamics as distinct from kinetic theory or statistical mechanics.
Historically, thermodynamic temperature was defined by Kelvin in terms of a macroscopic relation between thermodynamic wor ...
* is the gas constant
The molar gas constant (also known as the gas constant, universal gas constant, or ideal gas constant) is denoted by the symbol or . It is the molar equivalent to the Boltzmann constant, expressed in units of energy per temperature increment per ...
, which must be expressed in units consistent with those chosen for pressure, volume and temperature. For example, in SI units
The International System of Units, known by the international abbreviation SI in all languages and sometimes Pleonasm#Acronyms and initialisms, pleonastically as the SI system, is the modern form of the metric system and the world's most wid ...
= 8.3145 J⋅ K−1⋅ mol−1 when pressure is expressed in Pascals
The pascal (symbol: Pa) is the unit of pressure in the International System of Units (SI), and is also used to quantify internal pressure, stress, Young's modulus, and ultimate tensile strength. The unit, named after Blaise Pascal, is defined ...
, volume in cubic meters
The metre (British spelling) or meter (American spelling; see spelling differences) (from the French unit , from the Greek noun , "measure"), symbol m, is the primary unit of length in the International System of Units (SI), though its prefi ...
, and absolute temperature in Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and phys ...
.
The ideal gas law is an extension of experimentally discovered gas laws
The gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between pressure, volume and temperature of a sample of gas could be obtained which would hold to approximation for all gases.
Boyle ...
. It can also be derived from microscopic considerations.
Real fluid
In physics, a fluid is a liquid, gas, or other material that continuously deforms (''flows'') under an applied shear stress, or external force. They have zero shear modulus, or, in simpler terms, are substances which cannot resist any shear ...
s at low density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
and high temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
approximate the behavior of a classical ideal gas. However, at lower temperatures or a higher density, a real fluid deviates strongly from the behavior of an ideal gas, particularly as it condenses
Condensation is the change of the state of matter from the gas phase into the liquid phase, and is the reverse of vaporization. The word most often refers to the water cycle. It can also be defined as the change in the state of water vapor to ...
from a gas into a liquid or as it deposits
A deposit account is a bank account maintained by a financial institution in which a customer can deposit and withdraw money. Deposit accounts can be savings accounts, Transaction account#Current accounts, current accounts or any of several othe ...
from a gas into a solid. This deviation is expressed as a compressibility factor
In thermodynamics, the compressibility factor (Z), also known as the compression factor or the gas deviation factor, describes the deviation of a real gas from ideal gas behaviour. It is simply defined as the ratio of the molar volume of a gas to ...
.
This equation is derived from
* Boyle's law
Boyle's law, also referred to as the Boyle–Mariotte law, or Mariotte's law (especially in France), is an experimental gas law that describes the relationship between pressure and volume of a confined gas. Boyle's law has been stated as:
The ...
: ;
* Charles's law
Charles's law (also known as the law of volumes) is an experimental gas law that describes how gases tend to expand when heated. A modern statement of Charles's law is:
When the pressure on a sample of a dry gas is held constant, the Kelvin t ...
: ;
* Avogadro's law
Avogadro's law (sometimes referred to as Avogadro's hypothesis or Avogadro's principle) or Avogadro-Ampère's hypothesis is an experimental gas law relating the volume of a gas to the amount of substance of gas present. The law is a specific c ...
: .
After combining three laws we get
:
That is:
:
:.
Internal energy
The other equation of state of an ideal gas must express Joule's second law, that the internal energy of a fixed mass of ideal gas is a function only of its temperature, with . For the present purposes it is convenient to postulate an exemplary version of this law by writing:
:
where
* is the internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
* is the dimensionless specific heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
at constant volume, approximately for a monatomic gas
In physics and chemistry, "monatomic" is a combination of the words "mono" and "atomic", and means "single atom". It is usually applied to gases: a monatomic gas is a gas in which atoms are not bound to each other. Examples at standard conditions ...
, for diatomic
Diatomic molecules () are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen () or oxygen (), then it is said to be homonuclear. Ot ...
gas, and 3 for non-linear molecules if we treat translations and rotations classically and ignore quantum vibrational contribution and electronic excitation. These formulas arise from application of the classical equipartition theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
to the translational and rotational degrees of freedom.
That for an ideal gas depends only on temperature is a consequence of the ideal gas law, although in the general case depends on temperature and an integral is needed to compute .
Microscopic model
In order to switch from macroscopic quantities (left hand side of the following equation) to microscopic ones (right hand side), we use
:
where
* is the number of gas particles
* is the Boltzmann constant
The Boltzmann constant ( or ) is the proportionality factor that relates the average relative kinetic energy of particles in a gas with the thermodynamic temperature of the gas. It occurs in the definitions of the kelvin and the gas constant, ...
().
The probability distribution of particles by velocity or energy is given by the Maxwell speed distribution
Maxwell may refer to:
People
* Maxwell (surname), including a list of people and fictional characters with the name
** James Clerk Maxwell, mathematician and physicist
* Justice Maxwell (disambiguation)
* Maxwell baronets, in the Baronetage of ...
.
The ideal gas model depends on the following assumptions:
* The molecules of the gas are indistinguishable, small, hard spheres
* All collisions are elastic and all motion is frictionless (no energy loss in motion or collision)
* Newton's laws apply
* The average distance between molecules is much larger than the size of the molecules
* The molecules are constantly moving in random directions with a distribution of speeds
* There are no attractive or repulsive forces between the molecules apart from those that determine their point-like collisions
* The only forces between the gas molecules and the surroundings are those that determine the point-like collisions of the molecules with the walls
* In the simplest case, there are no long-range forces between the molecules of the gas and the surroundings.
The assumption of spherical particles is necessary so that there are no rotational modes allowed, unlike in a diatomic gas. The following three assumptions are very related: molecules are hard, collisions are elastic, and there are no inter-molecular forces. The assumption that the space between particles is much larger than the particles themselves is of paramount importance, and explains why the ideal gas approximation fails at high pressures.
Heat capacity
The dimensionless heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
at constant volume is generally defined by
:
where is the entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
. This quantity is generally a function of temperature due to intermolecular and intramolecular forces, but for moderate temperatures it is approximately constant. Specifically, the Equipartition Theorem
In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. T ...
predicts that the constant for a monatomic gas is = while for a diatomic gas it is = if vibrations are neglected (which is often an excellent approximation). Since the heat capacity depends on the atomic or molecular nature of the gas, macroscopic measurements on heat capacity provide useful information on the microscopic structure of the molecules.
The dimensionless heat capacity at constant pressure of an ideal gas is:
:
where is the enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
of the gas.
Sometimes, a distinction is made between an ideal gas, where and could vary with temperature, and a perfect gas
In physics and engineering, a perfect gas is a theoretical gas model that differs from real gases in specific ways that makes certain calculations easier to handle. In all perfect gas models, intermolecular forces are neglected. This means that one ...
, for which this is not the case.
The ratio of the constant volume and constant pressure heat capacity is the adiabatic index
In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure () to heat capacity at constant volu ...
:
For air, which is a mixture of gases, this ratio can be assumed to be 1.4 with only a small error over a wide temperature range.
Entropy
Using the results of thermodynamics
Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, entropy, and the physical properties of matter and radiation. The behavior of these quantities is governed by the four laws of the ...
only, we can go a long way in determining the expression for the entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodynam ...
of an ideal gas. This is an important step since, according to the theory of thermodynamic potentials, if we can express the entropy as a function of ( is a thermodynamic potential), volume and the number of particles , then we will have a complete statement of the thermodynamic behavior of the ideal gas. We will be able to derive both the ideal gas law and the expression for internal energy from it.
Since the entropy is an exact differential
In multivariate calculus, a differential or differential form is said to be exact or perfect (''exact differential''), as contrasted with an inexact differential, if it is equal to the general differential dQ for some differentiable function&nbs ...
, using the chain rule
In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
, the change in entropy when going from a reference state 0 to some other state with entropy may be written as where:
:
where the reference variables may be functions of the number of particles
The particle number (or number of particles) of a thermodynamic system, conventionally indicated with the letter ''N'', is the number of constituent particles in that system. The particle number is a fundamental parameter in thermodynamics which is ...
. Using the definition of the heat capacity
Heat capacity or thermal capacity is a physical property of matter, defined as the amount of heat to be supplied to an object to produce a unit change in its temperature. The SI unit of heat capacity is joule per kelvin (J/K).
Heat capacity i ...
at constant volume for the first differential and the appropriate Maxwell relation
file:Thermodynamic map.svg, 400px, Flow chart showing the paths between the Maxwell relations. P is pressure, T temperature, V volume, S entropy, \alpha coefficient of thermal expansion, \kappa compressibility, C_V heat capacity at constant volu ...
for the second we have:
:
Expressing in terms of as developed in the above section, differentiating the ideal gas equation of state, and integrating yields:
:
which implies that the entropy may be expressed as:
:
where all constants have been incorporated into the logarithm as which is some function of the particle number having the same dimensions as in order that the argument of the logarithm be dimensionless. We now impose the constraint that the entropy be extensive. This will mean that when the extensive parameters ( and ) are multiplied by a constant, the entropy will be multiplied by the same constant. Mathematically:
:
From this we find an equation for the function
:
Differentiating this with respect to , setting equal to 1, and then solving the differential equation yields :
:
where may vary for different gases, but will be independent of the thermodynamic state of the gas. It will have the dimensions of . Substituting into the equation for the entropy:
:
and using the expression for the internal energy of an ideal gas, the entropy may be written:
: