Physical Chemistry , 1st ed.

We can rearrange this. Bringing one term to the other side of the equation, we get



T

p



V



V

p



T



V

T



p

Multiplying everything to one side yields



T

p



V



V

p



T



V

T



p

 1 (1.25)

This is the cyclic rule for partial derivatives. Notice that each term involves p,

V, and T. This expression is independent of the equation of state. Knowing any

two derivatives, one can use equation 1.25 to determine the third, no matter

what the equation of state of the gaseous system is.

The cyclic rule is sometimes rewritten in a different form that may be eas-

ier to remember, by bringing two of the three terms to one side of the equa-

tion and expressing the equality in fractional form by taking the reciprocal of

one partial derivative. One way to write it would be



T

p



V

 (1.26)

This might look more complicated, but consider the mnemonic in Figure 1.11.

There is a systematic way of constructing the fractional form of the cyclic rule

that might be useful. The mnemonic in Figure 1.11 works for any partial de-

rivative in terms ofp,V, and T.

Example 1.7

Given the expression



T

p



V, n



V

p



T, n



V

T



p,n

determine an expression for



V

p



T, n

Solution

There is an expression involving Vand pat constant Tand non the right side

of the equality, but it is written as the reciprocal of the desired expression.

First, we can take the reciprocal of the entire expression to get



T

p



V, n

^

V

p



T, n



V

T

p,n

Next, in order to solve for ( V/ p)T, n, we can bring the other partial deriva-

tive to the other side of the equation, using the normal rules of algebra for

fractions. Moving the negative sign as well, we get



T

p



V, n



V

T



p,n



V

p



T, n

which provides us with the necessary expression.



V

T



p





V

p



T

1.7 More on Derivatives 19

 

Denominator

Numerator

p

V

()T

T

p

()T V V

()p

Figure 1.11 A mnemonic for remembering
the fraction form of the cyclic rule. The arrows
show the ordering of the variables in each partial
derivative in the numerator and denominator.
The only other thing to remember to include in
the expression is the negative sign.