Physical Chemistry , 1st ed.

Example 1.2

Calculate the volume of 1 mole of an ideal gas at SATP.

Solution

Using the ideal gas law and the appropriate value for R:

V

nR

P

T



V22.71 L

This is slightly larger than the commonly used molar volume of a gas at STP

(about 22.4 L), since the pressure is slightly lower.

1.5 Partial Derivatives and Gas Laws

A major use of equations of state in thermodynamics is to determine how one

state variable is affected when another state variable changes. In order to do

this, we need the tools of calculus. For example, a straight line, as in Figure

1.5a, has a slope given by y/x, which in words is simply “the change in yas

xchanges.” For a straight line, the slope is the same everywhere on the line.

For curved lines, as shown in Figure 1.5b, the slope is constantly changing.

Instead of writing the slope of the curved line as y/x, we use the symbol-

ism of calculus and write it as dy/dx, and we call this “the derivative ofywith

respect to x.”

Equations of state deal with many variables. The total derivativeof a func-

tion of multiple variables,F(x,y,z,.. .), is defined as

dF^

F

x



y,z,...

dx^

F

y



x,z,...

dy^

F

z



x,y,...

dz   (1.12)

In equation 1.12, we are taking the derivative of the function Fwith respect to

one variable at a time. In each case, the other variables are held constant. Thus,

in the first term, the derivative



^

F

x



y,z,... (1.13)

is the derivative of the function Fwith respect to xonly, and the variables y,z,

and so on are treated as constants. Such a derivative is a partial derivative.

The total derivative of a multivariable function is the sum of all of its partial

(1 mol)(0.08314 mLoblaKr)(273.15 K)



1 bar

8 CHAPTER 1 Gases and the Zeroth Law of Thermodynamics

y

x

(b)

dy

Slope dx

y

x

(a)

y

Slope  m x

y  mx  by  F(x)

dy

Slope dx

dy

Slope dx

Figure 1.5 (a) Definition of slope for a straight line. The slope is the same at every point on

the line. (b) A curved line also has a slope, but it changes from point to point. The slope of the

line at any particular point is determined by the derivative of the equation for the line.