Question

A monatomic ideal gas in a container of fixed volume contains 4.10 × 10^24 atoms.

a) If you add enough thermal energy to double the root-mean-square speed of the gas atoms, by how much have you changed the entropy of the gas?

Answer 1

given monoatomic gas

ideal gas in a container

fixed volume, V

n = 4.1*10^24 atoms

a. rms speed is given by

Vrms = sqrt(3RT/Mm)

Mm is molar mass of the gas

so enough energy is added to double the RMS speed

hence, temperature increases by 4 times

hence internal energy increases 4 times

let initial internal energy be U

then final internal energy = 4U

also, for a constant volume process

dS = integral(dU/T) = integral(n*CvdT/T*Ao)

where Ao is avogadro's number

hence

dS = n*Cv*ln(Tf/Ti)/Ao

Tf/Ti = 4

hence

Cv = 3R/2 ( monoatomic gas)

dS = 107.022134088 J/K