In
chemistry
Chemistry is the science, scientific study of the properties and behavior of matter. It is a natural science that covers the Chemical element, elements that make up matter to the chemical compound, compounds made of atoms, molecules and ions ...
, a steady state is a situation in which all
state variables
A state variable is one of the set of variables that are used to describe the mathematical "state" of a dynamical system. Intuitively, the state of a system describes enough about the system to determine its future behaviour in the absence of a ...
are constant in spite of ongoing
processes that strive to change them. For an entire system to be at
steady state
In systems theory, a system or a Process theory, process is in a steady state if the variables (called state variables) which define the behavior of the system or the process are unchanging in time. In continuous time, this means that for those p ...
, i.e. for all state variables of a system to be constant, there must be a flow through the system (compare
mass balance
In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have bee ...
). A simple example of such a system is the case of a bathtub with the tap running but with the drain unplugged: after a certain time, the water flows in and out at the same rate, so the water level (the state variable Volume) stabilizes and the system is in a steady state.
The steady state concept is different from
chemical equilibrium
In a chemical reaction, chemical equilibrium is the state in which both the reactants and products are present in concentrations which have no further tendency to change with time, so that there is no observable change in the properties of the sy ...
. Although both may create a situation where a
concentration
In chemistry, concentration is the abundance of a constituent divided by the total volume of a mixture. Several types of mathematical description can be distinguished: '' mass concentration'', ''molar concentration'', ''number concentration'', an ...
does not change, in a system at chemical equilibrium, the net
reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...
is zero (
products
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
transform into
reactant
In chemistry, a reagent ( ) or analytical reagent is a substance or compound added to a system to cause a chemical reaction, or test if one occurs. The terms ''reactant'' and ''reagent'' are often used interchangeably, but reactant specifies a ...
s at the same rate as reactants transform into products), while no such limitation exists in the steady state concept. Indeed, there does not have to be a
reaction
Reaction may refer to a process or to a response to an action, event, or exposure:
Physics and chemistry
*Chemical reaction
*Nuclear reaction
*Reaction (physics), as defined by Newton's third law
*Chain reaction (disambiguation).
Biology and me ...
at all for a steady state to develop.
The term steady state is also used to describe a situation where some, but not all, of the state variables of a system are constant. For such a steady state to develop, the system does not have to be a flow system. Therefore, such a steady state can develop in a closed system where a series of chemical reactions take place. Literature in
chemical kinetics
Chemical kinetics, also known as reaction kinetics, is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It is to be contrasted with chemical thermodynamics, which deals with the direction in wh ...
usually refers to this case, calling it ''steady state approximation''.
In simple systems the steady state is approached by state variables gradually decreasing or increasing until they reach their steady state value. In more complex systems state variables might fluctuate around the theoretical steady state either forever (a
limit cycle
In mathematics, in the study of dynamical systems with two-dimensional phase space, a limit cycle is a closed trajectory in phase space having the property that at least one other trajectory spirals into it either as time approaches infinity ...
) or gradually coming closer and closer. It theoretically takes an infinite time to reach steady state, just as it takes an infinite time to reach chemical equilibrium.
Both concepts are, however, frequently used
approximation
An approximation is anything that is intentionally similar but not exactly equality (mathematics), equal to something else.
Etymology and usage
The word ''approximation'' is derived from Latin ''approximatus'', from ''proximus'' meaning ''very ...
s because of the substantial mathematical simplifications these concepts offer. Whether or not these concepts can be used depends on the error the underlying assumptions introduce. So, even though a steady state, from a theoretical point of view, requires constant drivers (e.g. constant inflow rate and constant concentrations in the inflow), the error introduced by assuming steady state for a system with non-constant drivers may be negligible if the steady state is approached fast enough (relatively speaking).
Steady state approximation in chemical kinetics
The ''steady state approximation'', occasionally called the ''stationary-state approximation'' or
Bodenstein's ''quasi-steady state approximation'', involves setting the rate of change of a
reaction intermediate
In chemistry, a reaction intermediate or an intermediate is a molecular entity that is formed from the reactants (or preceding intermediates) but is consumed in further reactions in stepwise chemical reactions that contain multiple elementary st ...
in a
reaction mechanism
In chemistry, a reaction mechanism is the step by step sequence of elementary reactions by which overall chemical change occurs.
A chemical mechanism is a theoretical conjecture that tries to describe in detail what takes place at each stage of ...
equal to zero so that the kinetic equations can be simplified by setting the rate of formation of the intermediate equal to the rate of its destruction.
In practice it is sufficient that the rates of formation and destruction are ''approximately'' equal, which means that the net rate of variation of the concentration of the intermediate is small compared to the formation and destruction, and the concentration of the intermediate varies only slowly, similar to the reactants and products (see the equations and the green traces in the figures below).
Its use facilitates the resolution of the
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s that arise from
rate equation
In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reacti ...
s, which lack an
analytical solution
Generally speaking, analytic (from el, ἀναλυτικός, ''analytikos'') refers to the "having the ability to analyze" or "division into elements or principles".
Analytic or analytical can also have the following meanings:
Chemistry
* A ...
for most mechanisms beyond the simplest ones. The steady state approximation is applied, for example, in
Michaelis-Menten kinetics.
As an example, the steady state approximation will be applied to two consecutive, irreversible, homogeneous first order reactions in a closed system. (For
heterogeneous
Homogeneity and heterogeneity are concepts often used in the sciences and statistics relating to the uniformity of a substance or organism. A material or image that is homogeneous is uniform in composition or character (i.e. color, shape, siz ...
reactions, see
reactions on surfaces
Reactions on surfaces are reactions in which at least one of the steps of the reaction mechanism is the adsorption of one or more reactants. The mechanisms for these reactions, and the rate equations are of extreme importance for heterogeneous ca ...
.) This model corresponds, for example, to a series of
nuclear decompositions like .
If the rate constants for the following reaction are and ; , combining the
rate equation
In chemistry, the rate law or rate equation for a reaction is an equation that links the initial or forward reaction rate with the concentrations or pressures of the reactants and constant parameters (normally rate coefficients and partial reacti ...
s with a
mass balance
In physics, a mass balance, also called a material balance, is an application of conservation of mass to the analysis of physical systems. By accounting for material entering and leaving a system, mass flows can be identified which might have bee ...
for the system yields three coupled differential equations:
Reaction rates
For species A:
For species B: