Exercises

3.8 (a) The internal energy of a perfect monatomic gas relative to its value at T = 0 is . 32nRT

Calculate and for the gas.UV

T HV

T

Atkins says, that (3). The question gives us . dUUV

dVUT

dTT V= +

U nRT=

32

Holding T constant and rearranging this gives us . Since T is held

UV V

nRTT T =

32

constant, the right side of the equation becomes 0. UV

T = 0

The definition of enthalpy is . Replacing p with its Gas Law equivalent and H U pV= +taking the partial derivative with respect to V leaves us with the equation

. It has already been said that , and

HV

UV

nRTV

T T T =

+

UV

T

= 0

because T is held constant. That leaves .nRTV

T = 0HV

T = 0

Problems3.23 The speed of sound, , in a gas of molar mass M is related to the ratio of the heatcs

capacities by . Show that , where is the mass ( )c RT Ms = / /1 2 ( )c ps = / /1 2 density of the gas. Calculate the speed of sound in argon at 25 degrees C.

Rearranging the Gas Law for M gives us , and inserting this into the equation givenMRTp

=

gives us .cp

s =

1 2/

Variables

5/3R due to the fact that argon is monatomicR 8.31451 jk-1mol-1

M 39.95*10-3 Kgmol-1

T 298.5 K

Inserting this data into the equation , gives us the value 322 ms-1. This value( )c RT Ms = / /1 2is somewhat slower than the speed of sound in air, which is to be expected due to the increasedmolar mass of argon.