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Reversible Adiabatic Processes

An adiabatic process is one in which no heat is transferred to or from a closed system, so that dq is equal to zero for every infinitesimal step of the process:
Equation 16
Consider a reversible adiabatic process of an ideal gas.
Equation 17 and 18
Because dq = 0 for an adiabatic process, we equate dU and dw:
Equation 19
This is a differential equation that can be solved to give T as a function of V if the dependence of CV on T and V is known.
We first assume that CV is constant. We can solve Eq. (19) by separation of variables. We divide by T to separate the variables (remove any V dependence from the left-hand side and any T dependence from the right-hand side):
Equation 20
Because each integrand contains only one variable we can integrate Eq. (20) from the initial state, denoted by V1 and T1, to the final state, denoted by V2 and T2.A definite integration gives
We divide by CV and take the exponential (antilogarithm) of both sides of this equation:
Equation 21a
If the initial values V1 and T1 are specified, this equation gives T2 as a function of V2. If we drop the subscripts on V2 and T2, we can write Eq. (21a) in the form:
Equation 21b
Equation (21) can be used for a reversible adiabatic compression as well as for an expansion. It is an example of an important fact that holds for any system, not just an ideal gas: For a reversible adiabatic process in a simple system the final temperature is a function of the final volume for a given initial state. All of the possible final state points for reversible adiabatic processes starting at a given initial state lie on a single curve in the state space, called a reversible adiabat.
An equation analogous to Eq. (19) can be written for a real gas if CV is independent of V and if an adequate equation of state is available. For a gas obeying the van derWaals equation of state that
Equation 22
This can be used to derive an equation analogous to Eq. (21). For each such equation, there is a unique curve in the V–T plane containing all of the points that can be reached by adiabatic reversible processes from a given initial state.

Example 18

Show that for a reversible adiabatic process the van derWaals gas obeys
Equation 23


Consider a system containing 1.000 mol of gas. Using Eq. (22) we can write for a closed system
For a reversible process
For an adiabatic process
so that when terms are canceled

A reversible adiabatic process in a van derWaals gas with constant CV,m,
Equation 24
If the molar heat capacity of a van derWaals gas is represented by
Equation 25
the equation analogous to Eqs. (21) and (24) is
Equation 26
For a reversible adiabatic process, not only is T a function of V, but P is also a function of V. For an ideal gas with constant heat capacity, we can substitute the ideal gas equation, T  PV/nR, into Eq. (21) to obtain
Equation 27
The temperature can also be considered to be a function of the pressure:
Equation 28

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