An ideal gas of density 1.7 × 10⁻³ g cm⁻³ at a pressure of 1.5 × 10⁵ Pa is filled in a Kundt's tube. When the gas is resonated at a frequency of 3.0 kHz, nodes are formed at a separation of 6.0 cm. Calculate the molar heat capacities C_p and C_v of the gas.

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Solution

Given:
Density of the ideal gas, ρ = 1.7 × 10⁻³ g/cm³
= 1.7 k/gm³
Pressure of the gas, P = 1.5 × 10⁵ Pa
R = 8.3 J/mol-K

Resonance frequency of the gas = 3.0 kHz
Node separation in the Kundt's tube
= $\frac{l}{2}$ = 6 cm
So, l = 2 $\times$ 6 = 12 cm = 12 × 10⁻² m
So, V = fl = 3 × 10³ × 12 × 10⁻²
= 360 m/s
Speed of sound, V = $\sqrt{\frac{γ p}{ρ}}$
Or $V^{2} = \frac{γ p}{ρ}$
$∴ γ = \frac{v^{2} ρ}{P} = \frac{(360)^{2} \times 1.7 \times 10^{- 3}}{1.5 \times 10^{5}} = 1.4688 Using C_{p} - C_{v} = R a n d \frac{C_{p}}{C_{v}} = γ W e know that C_{v} = \frac{R}{γ - 1} = \frac{8.3}{0.4688} = 17.7 J / mol - K C_{p} = R + C_{v} = 8.3 + 17.7 = 26 J / mol - K$

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