Thermodynamics and Chemistry

CHAPTER 3 THE FIRST LAW

3.5 APPLICATIONS OFEXPANSIONWORK 75

On the molecular level, a gas with negligible intermolecular interactions^12 fulfills both
of these requirements. Kinetic-molecular theory predicts that a gas containing noninteract-
ing molecules obeys the ideal gas equation. If intermolecular forces (the only forces that
depend on intermolecular distance) are negligible, the internal energy is simply the sum of
the energies of the individual molecules. These energies are independent of volume but
depend on temperature.
The behavior of any real gas approaches ideal-gas behavior when the gas is expanded
isothermally. As the molar volumeVmbecomes large andpbecomes small, the average
distance between molecules becomes large, and intermolecular forces become negligible.

3.5.2 Reversible isothermal expansion of an ideal gas

During reversible expansion or compression, the temperature and pressure remain uniform.
If we substitutepDnRT=Vfrom the ideal gas equation into Eq.3.4.10and treatnandT
as constants, we obtain

wDnRT

ZV 2

V 1

dV

V

DnRTln

V 2

V 1

(3.5.1)

(reversible isothermal

expansion work, ideal gas)

In these expressions forwthe amountnappears as a constant for the process, so it is not
necessary to state as a condition of validity that the system is closed.

3.5.3 Reversible adiabatic expansion of an ideal gas

This section derives temperature-volume and pressure-volume relations when a fixed amount
of an ideal gas is expanded or compressed without heat.
First we need a relation between internal energy and temperature. Since the value of
the internal energy of a fixed amount of an ideal gas depends only on its temperature (Sec.
3.5.1), an infinitesimal change dTwill cause a change dUthat depends only onTand dT:

dUDf .T /dT (3.5.2)

wheref .T /D dU=dT is a function ofT. For a constant-volume process of a closed
system without work, we know from the first law that dUis equal to∂qand that∂q=dT
is equal toCV, the heat capacity at constant volume (Sec.3.1.5). Thus we can identify the
functionf .T /as the heat capacity at constant volume:

dUDCVdT (3.5.3)

(ideal gas, closed system)

The relation given by Eq.3.5.3is valid for any process of a closed system of an ideal gas
of uniform temperature, even if the volume is not constant or if the process is adiabatic,
because it is a general relation between state functions.

(^12) This book uses the terms “intermolecular interactions” and “intermolecular forces” for interactions or forces
between either multi-atom molecules or unbonded atoms.