Physical Chemistry , 1st ed.

Example 1.8

Use the cyclic rule to determine an alternate expression for



V

P



T

Solution

Using Figure 1.11, it should be easy to see that



V

p



T



You should verify that this is correct.

1.8 A Few Partial Derivatives Defined

Many times, gaseous systems are used to introduce thermodynamic concepts.

That’s because generally speaking, gaseous systems are well behaved. That is,

we have a good idea how they will change their state variables when a certain

state variable, controlled by us, is changed. Therefore gaseous systems are an

important part of our initial understanding of thermodynamics.

It is useful to define a few special partial derivatives in terms of the state

variables of gaseous systems, because the definitions either (a) can be consid-

ered as basic properties of the gas, or (b) will help simplify future equations.

The expansion coefficientof a gas, labeled , is defined as the change in vol-

ume as the temperature is varied at constant pressure. A 1/Vmultiplicative fac-

tor is included:



V

1



V

T



p

(1.27)

For an ideal gas, it is easy to show that R/pV.

The isothermal compressibilityof a gas, labeled , is the change in volume as

the pressure changes at constant temperature (the name of this coefficient is

more descriptive). It too has a 1/Vmultiplicative factor, but it is negative:



V

1



V

p



T

(1.28)

Because ( V/ p)Tis negative for gases, the minus sign in equation 1.28 makes

a positive number. Again for an ideal gas, it is easy to show that RT/p^2 V.

For both and , the 1/Vterm is included to make the quantities intensive

(that is, independent of amount*).

Since both of these definitions use p,V, and T, we can use the cyclic rule to

show that, for example,



T

p



V









T

p



V



V

T



p

20 CHAPTER 1 Gases and the Zeroth Law of Thermodynamics

*Recall that intensive properties (like temperature and density) are independent of

amount of material, whereas extensive properties (like mass and volume) are dependent on

the amount of material.