Find the number of degrees of freedom of molecules in a gas. Whose molar heat capacity at constant pressure $C_{P} = 29 J m o l^{- 1} K^{- 1}$

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Solution

Using $C_{P} - C_{V} = R$
$\Rightarrow C_{V} = C_{P} - R = 29 - 8.314 = 20.686 J m o l^{- 1} K^{- 1}$
$γ = \frac{C_{P}}{C_{V}} = \frac{29}{20.686} = 1.4$
using , $\frac{v_{r m s}}{v_{s o u n d}} = \sqrt{\frac{3}{γ}} = \sqrt{\frac{3 f}{f + 2}}$
or $\frac{3}{γ} = \frac{3 f}{f + 2}$
or $f + 2 = γ f = 1.4 f$
$∴ f = \frac{2}{0.4} = 5$

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C = 29 J m o l^{- 1} K^{- 1}

C_{P} = 29 J/(mol.K)

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C_{v} = 650 J K g^{- 1} K^{- 1}

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