A polyatomic ideal gas has 24 vibrational mod | vedclass

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(A) For a polyatomic gas,the total degrees of freedom $f$ is given by the sum of translational,rotational,and vibrational degrees of freedom.Translati

Vedclass · Accepted Answer

$1.03$

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(A) For a polyatomic gas,the total degrees of freedom $f$ is given by the sum of translational,rotational,and vibrational degrees of freedom.Translational degrees of freedom $= 3$.Rotational degrees of freedom for a non-linear polyatomic molecule $= 3$.Each vibrational mode contributes $2$ degrees of freedom (one for kinetic energy and one for potential energy).Given $24$ vibrational modes,vibrational degrees of freedom $= 24 	imes 2 = 48$.Total degrees of freedom $f = 3 + 3 + 48 = 54$.The adiabatic index $\gamma$ is given by $\gamma = 1 + \frac{2}{f}$.Substituting the value of $f$: $\gamma = 1 + \frac{2}{54} = 1 + \frac{1}{27} = \frac{28}{27} \approx 1.037$.Rounding to two decimal places,we get $\gamma = 1.03$.

List-$I$	List-$II$
$(A)$ $3$ Translational degrees of freedom	$(I)$ Monoatomic gases
$(B)$ $3$ Translational,$2$ rotational degrees of freedom	$(III)$ Rigid diatomic gases
$(C)$ $3$ Translational,$2$ rotational and $1$ vibrational degrees of freedom	$(IV)$ Non-rigid diatomic gases
$(D)$ $3$ Translational,$3$ rotational and more than one vibrational degrees of freedom	$(II)$ Polyatomic gases

$A$ polyatomic ideal gas has $24$ vibrational modes. What is the value of $\gamma$ ?

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