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An ideal gas consists of three dimensional polyatomic molecules. The temperature is such that only one vibrational mode is excited. If $R$ denotes the gas constant, then the specific heat at constant volume of one mole of the gas at this temperature is:

The method I used is that
$$C_v=\frac{f}{2}R$$ Where Cv = specific heat at constant volume f= degree of freedom

Given that it is a 3-D polyatomic molecules, it would have the following degrees of freedoms

a) 3 translational degree freedom

b) 3 Rotational degree of freedom

c) 1 vibrational degree of freedom ( given in question)

Total degree of freedoms are 7, hence
$$Cv=\frac{7}{2}R$$

The twist, however, is that the answer is $$Cv=4R$$

Can someone explain where the 8th degree of freedom came from? or Is it that the answer given is wrong?

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  • $\begingroup$ this question has been answered about two days back...appears to be duplicate..the wording is different. $\endgroup$
    – drvrm
    Feb 28 '18 at 13:15
  • $\begingroup$ Any link of the question? $\endgroup$ Feb 28 '18 at 13:20
  • $\begingroup$ searching for it.... $\endgroup$
    – drvrm
    Feb 28 '18 at 13:23
  • $\begingroup$ Hint: In vibrational mode - a vibration involves both kinetic and potential energy terms which are squares of velocity and coordinate thereby on the average it may contribute two degrees of freedom.. $\endgroup$
    – drvrm
    Feb 28 '18 at 13:27
  • $\begingroup$ If that is the case then how come diatomic molecules have 6 degrees of freedom ( 3 translational, 2 rotational and 1 vibrational) and not 7 degrees of freedom ( 3 translation, 2 rotational and 2 due to vibrations) $\endgroup$ Mar 1 '18 at 8:52
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Your formula for Cv is wrong. The formula u used is for that condition when Vibrational mode is absent. Actual in general formula is Cv= ( ft/2 + fr/2 + fv) R Now using these u will get the value as 4R.

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Cv=(3+f)R. As, f=1, Cv=4R Cp=(4+f)R

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The degrees of freedom for a 3D polyatomic gas molecule are $6$ at normal temperature. But as there is only one vibrational mode so the degrees of freedom become $6+2=8$. Hence $C_v=fR/2$ so $C_v=4R$. Here we add 2 because in a polyatomic gas molecule no.of vibrational mode is $3N-6$ (for non linear). So $3 \cdot 3-6=3$. But only one vibrational mode is active so degrees of freedom become $8$.

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  • $\begingroup$ Please use mathjax for formatting $\endgroup$
    – Semoi
    Jan 12 '20 at 11:11
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one vibrational mode contributes 2 degrees of freedom hence 3+3+2 = 8 then (8/2)R = 4R

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