Heat Capacity Difference & the Joule-Thomson Effect

Previously in this course we pulled from the thermodynamic heavens that, for an ideal gas, C_P - C_V = nR. Our first task in today's lecture was to find an expression for the heat capacity difference of any system and, in doing so, we showed from where the ideal gas relationship arises. Midstream in the derivation we paused to explain why C_P and C_V values are nearly identical for condensed phases (solids and liquids) under normal conditions, another result that was simply asserted before. Remember that the goal of pchem is to explain all of chemistry using simple models, so we like to pull relations from nowhere as rarely as possible.

In Wednesday's lecture, we had expanded on how the internal energy U varies with T and V, obtaining the internal pressure concept [(∂U/∂V)_T] in the process. Today we performed an analogous treatment of the enthalpy H, finding how it varies with T and P, making it the third time this quarter we've associated U with V and H with P. Engel-Reid does not use this terminology but our results (Equations 3.20, 3.44) are commonly called the thermodynamic equations of state. Note that these are different from plain-vanilla equations of state, which tie together physical properties of a system, like P, T and V. The derivative (∂H/∂P)_T will be rather important when we consider reactions and other processes that do not occur at 1 bar (for example, reactions occuring in the troposphere or several miles into the earth's mantle).

As we finish up Chapter 3, we then discussed the Joule-Thomson effect and coefficient, μ_T=(∂T/∂P)_H, which measure the temperature response of a substance (usually a gas) to changes in pressure at constant enthalpy. Our intuition suggests that gases cool upon expansion, which is usually true and can be seen explicitly from the ideal gas law. But real gases are often unpredictable and several (like hydrogen and helium) have negative JT coefficients at standard conditions, meaning that they increase in temperature upon expansion, which is particularly important when handling tanks of hydrogen gas. The throttling apparatus devised by Joule and Thomson to attain isenthalpic conditions (in which no heat is extracted from nor no net work is done on the system) is particularly clever.

Monday will bring us to the [hopefully] familiar topic of thermochemistry [Chapter 4, which we will finish on Wednesday] and quiz 2, with exam 1 looming just over the horizon.