Explanation:
The velocity of sound depends on the density of the medium. So we need to find the density of air at each set of conditions. The density of air is:
ρ = (Pd / (Rd T)) + (Pv / (Rv T))
where Pd and Pv are the partial pressures of dry air and water vapor,
Rd and Rv are the specific gas constants of dry air and water vapor,
and T is the absolute temperature.
At the first condition:
Pv = 31.7 mmHg = 4226.3 Pa
Pd = 650 mmHg - 31.7 mmHg = 618.3 mmHg = 82433 Pa
Rv = 461.52 J/kg/K
Rd = 287.00 J/kg/K
T = 30°C = 303.15°C
ρ = (82433 / 287.00 / 303.15) + (4226.3 / 461.52 / 303.15)
ρ = 0.94746 + 0.03021
ρ = 0.97767 kg/m³
At the second condition:
Pv = 0 Pa
Pd = 650 mmHg = 86660 Pa
Rv = 461.52 J/kg/K
Rd = 287.00 J/kg/K
T = 0°C = 273.15°C
ρ = (86660 / 287.00 / 273.15) + (0 / 461.52 / 273.15)
ρ = 1.1054 + 0
ρ = 1.1054 kg/m³
The square of the velocity of sound is proportional to the ratio between pressure and density:
v² = k P / ρ
Since the atmospheric pressure is constant, we can say it's proportional to just the density:
v² = k / ρ
Using the first condition to find the coefficient:
(340)² = k / 0.97767
k = 113018.652
Now finding the velocity of sound at the second condition:
v² = 113018.652 / 1.1054
v = 319.75
