Physical Chemistry Third Edition

2.3 Internal Energy: The First Law of Thermodynamics 59

wherekBis Boltzmann’s constant,nis the amount of gas in moles,Ris the ideal gas

constant, andT is the absolute temperature. Since this energy does not include the

gravitational potential energy and the kinetic energy of the entire system, this energy

is the internal energy of the model gas:

U

3

2

nRT (dilute monotonic gas) (2.3-7)

Real atoms and molecules are not structureless particles. Real atoms and molecules

have translational energy and electronic energy, and molecules also have rotational

and vibrational energy. The translational energy of all dilute gases is accurately

represented by Eq. (2.3-7). The other contributions to the energy of a dilute gas

are studied in statistical mechanics, which is discussed in later chapters of this

textbook. At ordinary temperatures the electrons of nearly all atoms and molecules

are in their lowest possible energy states, and the electronic energy is a constant,

which we can set equal to zero. The vibrational contribution to the energy is not

quite so small as the electronic contribution at ordinary temperatures, but we will

neglect it for now. Statistical mechanics gives the following results for the rotational

contributions:

UrotnRT (diatomic gas or linear polyatomic gas) (2.3-8)

Urot

3

2

nRT (nonlinear polyatomic gas) (2.3-9)

We can now write formulas for the internal energy of dilute gases:

U≈

3

2

nRT+U 0 (monatomic gas) (2.3-10)

U≈

5

2

nRT+U 0 (diatomic gas or linear polyatomic gas) (2.3-11)

U≈ 3 nRT+U 0 (nonlinear polyatomic gas) (2.3-12)

We can set the zero of energy at any convenient energy, so we ordinarily set the constants

denoted byU 0 equal to zero. We will later use experimental heat capacity data to test

these equations. Near room temperature, Eq. (2.3-10) is a very good approximation

and Eqs. (2.3-11) and (2.3-12) are fairly good approximations.

The atoms or molecules of a solid or liquid have the same average translational

energy as the molecules of a gas at the same temperature, although the translational

motion is a kind of rattling back instead of motion in a straight line between collisions.

In some liquids containing small molecules the rotational and vibrational contributions

will also be nearly the same, although in other cases the rotation is restricted. However,

the molecules of a liquid or solid are packed closely together, and the potential energy

makes an important contribution to the internal energy. There is no simple representa-

tion of this potential energy, and it is not possible to represent the internal energy of a

solid or liquid by a general mathematical formula.