PV Diagrams and Work Done in Constrained Processes | *Six Ideas That Shaped Physics* Chapter T7 Notes

A useful way to analyze the way a process changes the macrostate of a gas -- properties like $P, V, T, U, M, N$ -- is to graph the way a process changes the gas's pressure and volume. Such a graph is called a PV diagram.

In particular, there are a few types of constrained processes of note:

• Isochoric process: a gas is heated or cooled while its volume stays the same

• Isobaric process: a gas is heated or cooled while its pressure stays the same

• Isothermal process: a gas is expanded or compressed while its temperature is constrained to be constant

• Adiabatic process: a gas is expanded or compressed while heat flow is constrained to be zero

A PV diagram with each type of contrained process is shown below:

(figure T7.4)

Consider the work done for each type of process. The work done is expressed by the integral $- \int P dV$.

• For an isochoric process, the volume does not change, so $W = - \int P dV = 0$.

• For an isobaric process, $P$ is constant, so $W = - \int P dV = - P \int dV = -P (V_f - V_i) = -P \Delta V$.

• For an isothermal process, $T$ is constant, so the ideal gas law tells us that $P$ depends only on $V$: $PV=Nk_B T \implies P(V) = \frac{N k_B T}{V} = \frac{P_i V_i}{V}$ and thus $W = \N k_B T \ln \frac{V_f}{V_i} = - P_i V_i \ln \frac{V_f}{V_i}$

Usefully, the work done by a quasistatic (slow enough that the pressure and temperature of the gas remains the same throughout it at all times) expansion or compression is also equal to the area under the curve of the process plotted on the PV diagram. (question for self: why?)

(figure T7.7)

For an adiabatic process, the relation between changes in pressure, volume, and temperature are a little more complicated.

Since no heat is added, the work done $dW = -P dV$ is equal to the change in internal energy $dU = \frac{f}{2} N k_B dT$.

Then we can take the derivative $\frac{d}{dt}$ on both sides of the ideal gas law $PV = N k_B T \implies P \frac{dV}{dT} + V \frac{dP}{dT} = N k_B \implies P dV + V dP = N k_B dT$. Plugging in the earlier relation $-P dV = \frac{f}{2} N k_B dT$ we can simplify this equation:

$$P dV + V dP = N k_B dT \implies P dV + V dP = =\frac{2}{f} P dV \implies \gamma P dV + V dP = 0$$

with $\gamma = 1 + \frac{2}{f}$. Multiplying both sides by $\frac{V^{\gamma - 1}}{dV}$, we have:

$$\gamma PV^{\gamma - 1} + V^{\gamma} \frac{dP}{dV} \implies \frac{d}{dV} (PV^\gamma) = 0$$

Thus $PV^\gamma$ is not changed by an adiabatic process. Plugging the ideal gas law back in reveals that $\frac{d}{dV} (TV^{\gamma -1}) = 0$ as well and so $TV^{\gamma-1}$ is also constant. (note to self: actually derive this)

Constrained processes also result in different heat capacities for the gasses undergoing heating or cooling. For an isochoric process, where volume is held constant, there is no work done so $C = \frac{dQ}{dT} = \frac{dU}{dT} = \frac{f}{2}N k_B$. More often, gas processes happen under the constraint that pressure is held constant, for example in a flexible container under the constant pressure of the atmosphere. Under these conditions, some of the energy put into a gas goes towards expanding the volume of the container to maintain constant pressure, and thus the heat capacity is higher. Specifically $dU = dQ + dW \implies C = \frac{dQ}{dT} = \frac{dU}{dT} - \frac{dW}{dT} = \frac{f}{2} N k_B + N k_B = (1 + \frac{f}{2}) N k_B = \gamma \frac{f}{2} N k_B = \gamma C_V$.