The heat capacity, CV,m, of 1 mol of monoatomic gases such as helium at a constant-volume condition — Physical Chemistry — Thermodynamics Chemistry Question
Speed of sound
The heat capacity, CV,m, of 1 mol of monoatomic gases such as helium at a constant-volume condition is expressed by the following equation:
RCV 2 3
m, =
Here, R is the gas constant. The CV, m value corresponds to the increase in the energy of flying motion of gaseous atoms per unit temperature, and the flight speed of the atoms is expected to reduce to zero (0) at 0 K.
Derive the mean flight speed, v, of gaseous atoms with molar mass M at temperature T.
Model Answer
Since the flight energy is zero at 0 K and it increases by R 2 3 per unit temperature (per mol), the energy should be RT 2 3 (per mol) at temperature T.
Thus, the total kinetic energy of one mole of gas is RTMv 2 3
2 1 2 = .
This equation is solved for v:
M RT
v 3=
The speed of sound, vs, in monoatomic gases is proportional (and roughly equal) to the flight speed, v, of the gaseous atoms. The speeds of sound in He (helium) and Ar (argon) at room temperature are 1007 m s–1 and 319 m s–1, respectively.
Estimate the speed of sound in Ne (neon), vs(Ne), at room temperature.
Model Answer
From the solution of problem 2.1, v, and therefore, vs should be proportional to M–1/2. Therefore, the speed of sound in Ne can be estimated as
vs(Ne) = 1007 × (20.18/4.003)–1/2 = 448 m s–1
( ) 1 –1 s Ne = 1007 = 448 m s
20.18 4.003
v
or ( ) 1 –1 s Ne = 319 = 449 m s
20.18 39,95
v