So the exact question is

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

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?

  • $\begingroup$ 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? $\endgroup$ – Shikhar Asthana 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) $\endgroup$ – Shikhar Asthana Mar 1 '18 at 8:52

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


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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