Physical Chemistry Third Edition

62 2 Work, Heat, and Energy: The First Law of Thermodynamics

CV≈ 3 nR (nonlinear polyatomic dilute gas) (2.4-9)

This ascribes the internal energy to the translational energy and rotational energy of the

molecules. Equations (2.4-8) and (2.4-9) are usable approximations for some substances

near room temperature.

Amounts of Heat Transferred to an Ideal Gas

In an isothermal reversible expansion or compression of a closed ideal gas,∆Uvanishes

so that

qrev∆U−wrev−wrev

∫V 2

V 1

PdVnRT

∫V 2

V 1

1

V

dV

qrevnRTln

(

V 2

V 1

)

(ideal gas; reversible

isothermal change)

(2.4-10)

whereV 2 is the final volume andV 1 is the initial volume. We will apply this equation

to dilute gases, but must remember not to apply it to other systems.

EXAMPLE2.14

Find the amount of heat put into 5.000 mol of argon (assumed ideal) in expanding reversibly

and isothermally at 298.15 K from a volume of 20.00 L to 100.00 L.

Solution

w−(5.000 mol)(8.3145 J K−^1 mol−^1 )(298.15 K) ln

(

100 .0L

20 .00 L

)

−19950 J

q∆U−w−w19950 J

Exercise 2.8

Calculate the amount of heat put into the system of the previous example if it expands irreversibly

and isothermally at 298.15 K at a constant external pressure of 1.000 atm from a volume of 20.00 L

to a volume of 100.00 L. Assume thatP(transmitted)Pext.Hint:∆Uis the same as in the

example becauseUis a state function.

SinceUdepends only onTandnfor an ideal gas,

(

∂U

∂V

)

T,n

0 (ideal gas) (2.4-11)

For an ideal gas with fixedn(a closed system)

dUCVdT (closed ideal gas) (2.4-12)