Understanding the Speed of Sound
The speed of sound in a gas is primarily determined by:
* Temperature (T): Higher temperatures mean faster-moving molecules, leading to a faster transmission of sound waves.
* Molar mass (M): Lighter molecules travel faster, so gases with lower molar masses have faster sound speeds.
The Formula
The speed of sound (v) in an ideal gas is given by:
v = √(γRT/M)
Where:
* γ (gamma) is the adiabatic index (ratio of specific heats) - approximately 1.4 for diatomic gases like oxygen and 1.66 for monatomic gases like helium.
* R is the ideal gas constant.
* T is the absolute temperature (in Kelvin).
* M is the molar mass of the gas.
Calculating the Ratio
1. Set up the ratio:
(v_He / v_O2) = √[(γ_He * R * T / M_He) / (γ_O2 * R * T / M_O2)]
2. Simplify:
(v_He / v_O2) = √[(γ_He * M_O2) / (γ_O2 * M_He)]
3. Plug in values:
* γ_He ≈ 1.66
* γ_O2 ≈ 1.4
* M_He ≈ 4 g/mol
* M_O2 ≈ 32 g/mol
(v_He / v_O2) = √[(1.66 * 32) / (1.4 * 4)] ≈ √(9.43) ≈ 3.07
Result:
The ratio of the speed of sound in helium to that in oxygen at the same temperature and pressure is approximately 3.07. This means sound travels about three times faster in helium than in oxygen.
