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:

Consider a reversible adiabatic process of an ideal gas.

Because dq = 0 for an adiabatic process, we equate dU and dw:

This is a *differential equation* that can be solved to give T as a function of V if the dependence of C_{V} on T and V is known.

We first assume that C_{V} 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):

Because each integrand contains only one variable we can integrate Eq. (20) from the initial state, denoted by V_{1} and T_{1}, to the final state, denoted by V_{2} and T_{2}.A definite integration gives

We divide by C_{V} and take the exponential (antilogarithm) of both sides of this equation:

If the initial values V_{1} and T_{1} are specified, this equation gives T_{2} as a function of V_{2}. If we drop the subscripts on V_{2} and T_{2}, we can write Eq. (21a) in the form:

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 C_{V} is independent of V and if an adequate equation of state is available. For a gas obeying the van derWaals equation of state that

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

Solution

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 C_{V,m},

If the molar heat capacity of a van derWaals gas is represented by

the equation analogous to Eqs. (21) and (24) is

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

The temperature can also be considered to be a function of the pressure: