An ideal gas having density $1.7 \times 10^{- 3} g c m^{- 3}$ at a pressure $1.5 \times 10^{5}$ Pa is filled in a Kundt 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 capaticies $C_{p}$ and $C_{v}$ of the gas.

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Solution

P = $1.7 \times 10^{- 3} g / c m^{3}$

= $1.7 k / g m^{3}$

P = $1.5 \times 10^{5}$ Pa

R = 8.3 J/mol 1-K,

F= 3.0 KHz

Node separation in a Kundt's tube

= $\frac{1}{2} = 6 c m$

l = 12 cm = 12 $\times 10^{- 2} m$

So, V = ft = 3 $\times 10^{3} \times 12 \times 10^{- 2} m$

= 360 m/s

We know, speed of sound = $\sqrt{\frac{γ p}{p}}$

or $V^{2} = \frac{γ p}{p}$

$∴ γ = \frac{v^{2} p}{p} = \frac{(360)^{2} 1.7}{1.5 \times 10^{5}}$

= 1.4688

But $C_{v} = \frac{R}{γ - 1} = \frac{8.3}{0.4688}$

= 17.71 J/mol-K

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