# Internal energy of any ideal gas

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• Pushoam

#### Pushoam

The internal energy of monoatomic ideal gas is due to the kinetic energy of the molecules.
Using Boltzmann Maxwell distribution, it is calculated that the kinetic energy due to translational motion of gas molecules of an ideal gas depends only on the temperature.
In case of monoatomic gas, since the molecules can have only transational motion, the internal energy depends only on the temperature.

But, it is said that the internal energy of any ideal gas depends upon only on the temperature. Is it an approximation?

Multiatomic gasses also have rotational kinetic energy and vibrational energy.
That is, they have more degrees of freedom.
Otherwise the same things apply.
It means that the internal energy indeed depends only on temperature - just with a different constant.
$$U=n C_{V,m}T$$
where ##n## is the number of moles, ##C_{V,m}=\frac f2 R## is the molar heat capacity, and ##f## is the number of degrees of freedom (f=3 for monatomic ideal gasses, f=5 for diatomic ideal gasses).
For real gasses it's of course a little more complicated.
See for instance Heat Capacity on wiki.

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Thank you.