Physical Chemistry Third Edition

58 2 Work, Heat, and Energy: The First Law of Thermodynamics

Exercise 2.6

a.Assume that argon is an ideal gas withCV,m 12 .47JK−^1 mol−^1. Find∆Uif 1.000 mol of

argon gas is heated at constant volume from 298.15 K to 500.0 K. Find the ratio of this energy

difference to the rest-mass energy of the system. Find the difference between the observed

mass of the system at 298.15 K and at 500.0 K.

b.Explain why it would be difficult to use values of total energies for chemical purposes if the

rest-mass energy were included.

For a one-phase simple system containing one component the equilibrium state is

specified by the values of three variables, at least one of which must be extensive. Since

the internal energy is a state function we can write

UU(T,V,n) (2.3-4)

or

UU(T,P,n) (2.3-5)

The internal energy is a state function, but heat and work are not state functions.

Because heat and work are both means of changing the value of the internal energy, they

do not maintain separate identities after a transfer of energy is finished. The following

analogy has been used.^3 Heat transferred to the system is analogous to rain falling on a

pond, work done on the system is analogous to the influx of a stream into the pond, and

energy is analogous to water in the pond. Evaporation (counted as negative rainfall)

is analogous to heat flow to the surroundings, and efflux from the pond into a second

stream is analogous to work done on the surroundings. Once rain falls into the pond,

it is no longer identifiable as rain, but only as water. Once stream flow is in the pond,

it also is identifiable only as water, and not as stream flow. The amount of water in the

pond is a well-defined quantity (a state function), but one cannot separately state how

much rain and how much stream flow are in the pond. Similarly, there is no such thing

as the heat content of a system in a given state and no such thing as the work content

of a system in a given state.

The Molecular Interpretation of the Internal Energy

Although the internal energyUis a macroscopic state function, it must include the

energies of the atoms or molecules making up the system. In Chapter 9 we will discuss

a model system that represents a dilute monatomic gas. In this model, the molecules

are represented by structureless mass points with no intermolecular forces. The con-

stant potential energy is set equal to zero. The only energy is the translational kinetic

energy of the molecules, and it will be found that the energy of the model gas ofN

molecules is

E

3

2

NkBT

3

2

nRT (dilute monotonic gas) (2.3-6)

(^3) Herbert B. Callen,Thermodynamics, Wiley, New York, 1960, p. 19.