Concise Physical Chemistry

c03 JWBS043-Rogers September 13, 2010 11:24 Printer Name: Yet to Come

50 THE THERMODYNAMICS OF SIMPLE SYSTEMS

Cp

T

FIGURE 3.6 Typical heat capacity as a function of temperature for a simple organic

molecule. Allowed modes of motion are gradually activated as the gas is warmed and more

thermal energy becomes available. (See Klotz and Rosenberg, 2008 for more detail.)

degrees of freedom, one for each possible mode of motion. As in the hydrogen case,

not all modes of motion are activated. At any given temperature, a molecule may

have many degrees of freedom available to it but not enough thermal energy to fully

activate all modes. For this reason, heat capacity curves are sigmoidal (S-shaped)

starting from zero at 0 K, where there is no motion at all, and rising gradually, as

modes of motion are activated, to a limiting value determined by how complicated

the molecule is.

3.11 ADIABATIC WORK

Because partial derivatives like(∂U/∂T)p, and so on, can be handled just as though

they were algebraic variables, it is possible to develop quite an arsenal of equa-

tions relating the first law quantities described so far and to expand them to include

other variables (Klotz and Rosenberg, 2008). An important concept is that ofadi-

abatic(perfectly insulated) work done on or by a gas. The workdwbehaves like

a thermodynamic function because the path has been specified by settingq=0.

NowdU=dq+dw=pdVfor a system restricted to pressure–volume work. The

energyUis a state variableU=f(V,T) for one mole, so

dU=

(

∂U

∂V

)

T

dV+

(

∂U

∂T

)

V

dT

and

(

∂U

∂V

)

T

dV+

(

∂U

∂T

)

V

dT+pdV= 0

The first term above drops out if we consider expansion of an ideal gas because the

functional dependence onVdisappears (Joule experiment). Also,(∂U/∂T)V=CV