11.3.5 Internal Energy Relationship with Temperature

In this section, we will discover the quantitative link between a gas’s internal energy and its temperature. We start with the equation of state ( $\displaystyle pV=nRT$ ) and the pressure formula ( $\displaystyle \displaystyle p=\frac{1}{3}\rho \left\langle {{{c}^{2}}} \right\rangle$ ), written in their alternative form.

$\displaystyle pV=NkT$

$\displaystyle \displaystyle pV=\frac{1}{3}Nm\left\langle {{{c}^{2}}} \right\rangle$

Combining them, we obtain

$\displaystyle \begin{aligned}\frac{1}{3}Nm\left\langle {{{c}^{2}}} \right\rangle &=NkT\\m\left\langle {{{c}^{2}}} \right\rangle &=3kT\\\frac{1}{2}m\left\langle {{{c}^{2}}} \right\rangle &=\frac{3}{2}kT\\\left\langle {\frac{1}{2}m{{c}^{2}}} \right\rangle &=\frac{3}{2}kT\end{aligned}$

Recognize that $\displaystyle \displaystyle \left\langle {\frac{1}{2}m{{c}^{2}}} \right\rangle$ represents the average translational KE of gas particles in the gas. So

$\displaystyle \displaystyle \left\langle {KE} \right\rangle =\frac{3}{2}kT$

Voila. We now have an understanding of temperature at a molecular level. The thermodynamic temperature is directly proportional to the average (translational) kinetic energy of the gas particles!

Remember that the internal energy $\displaystyle U=P{{E}_{{total}}}+K{{E}_{{total}}}$ ? An ideal gas does not have any potential energy. So for an ideal gas, U can be simplified to be[1]

$\displaystyle \displaystyle U=K{{E}_{{total}}}=\frac{3}{2}NkT=\frac{3}{2}nRT$

It turns out that the internal energy of an ideal gas depends only on its temperature. Or in mathematical jargon, we say that U is a function of T only.

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Video Explanation

Derivation of <KE>=3/2kT

Concept Test

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[1] Actually, $ $\displaystyle \displaystyle U=\frac{3}{2}nRT$ is valid only if the gas is ideal (if not PE_total won’t be zero) and monatomic (if not KE_total will include some rotational KE in addition to translational KE). If you’re interested, go read section 11.A.

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11.3.5 Internal Energy Relationship with Temperature

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