Question

4.0 g of helium occupies 22400 cm³ at STP. The specific heat capacity of helium at constant pressure is 5.0 cal K⁻¹ mol⁻¹. Calculate the speed of sound in helium at STP.

Answer 1

Given:
Specific heat capacity at constant pressure, C_p = 5.0 cal/mol-K
C_p = 5.0 × 4.2 J/mol-K
C_p = 21 J/mol-K
Volume of helium, V = 22400 ${\mathrm{cm}}^3$ = 0.0224 ${\mathrm{m}}^3$
At STP, P = 1 atm = ${10}^5\mathrm{Pa}$
The speed of sound in gas,
$$\begin{gathered} v\mathit{=}\sqrt{\frac{\mathit{\gamma }\mathit{p}}{\mathit{\rho }}}\mathit{=}\sqrt{\frac{\mathit{\gamma }\mathit{R}\mathit{T}}{\mathit{M}}}\mathit{=}\sqrt{\frac{\mathit{\gamma }\mathit{P}\mathit{V}}{\mathit{M}}} \\ {C}_{\mathit{p}}\mathit{=}\frac{\mathit{R}\mathit{\gamma }}{\mathit{\gamma }\mathit{-}\mathit{1}} \end{gathered}$$
Or 21(γ − 1) = 8.3γ
⇒ 21γ − 8.3γ = 21
⇒ 12.7γ = 21
$$\begin{gathered} \therefore \gamma =\frac{21}{12.7}=1.653 \\ v=\sqrt{\frac{1.653\times 1.0\times {10}^5\times 0.0224}{4\times {10}^{-3}}} \\ v=960\mathrm{m}/\mathrm{s} \end{gathered}$$