A particle gas consists of N monatomic particles each of mass m all contained in a volume V at temperature T. Your answers should be written in terms of the Boltzmann constant kB and Avagadro's number NArather than R=NAkB.

A) Express the pressure p of the gas in terms of its energy density U/V.
Enter the numerical factor that multiplies U/V in your expression for p to at least three significant figures.
B) What will be the ratio of the new molar mass
M? to the old molar mass M?
C) What will be the ratio of the new rms speed v?rms to the old rms speed vrms?
D) What will be the ratio of the new molar heat capacity C?V to the old molar heat capacity CV?

a)p = 0.67 U/V

b)M´/M = 3

c)v´rms/vrms = 0.58

d)C´v/Cv = 1

Explanation:

a) An equation must be found that relates the pressure to the volume of the gas. Then we will use these equations for U and write p in relation to U/V. According to the ideal gas law, we have:

pV = NkbT

clearing p:

p = (Nkb)*T/V = 0.67 U/V

b) the ratio of the new molar mass M´ to the old molar mass M would be equal to:

M´/M = 3

c) According to the equation vrms = (<v^2>)^1/2. Monatomic particle has three degrees of freedom. The kinetic energy associated with each degree of freedom is equal to K = (1/2)*kB*T. Another equation for kinetic energy is equal to K=(1/2)*m*<vx^2>

v´rms/vrms = 0.58

d) The total heat capacity of a gas can be calculated by multiplying the heat capacity per mole by the number of moles of the gas particles n=N/NA

C´v/Cv = 1

see earlier