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Mean kinetic energy of one mole of an ideal gas

The mean kinetic energy of one mole of gas is equivalent to the mean kinetic energy of one gas particle multiplied by Avogadro's constant $$N_{\text{A}}$$. This is given by:$$$\frac{1}{2}mN_{\text{A}}\langle v^{2}\rangle=\frac{3}{2}kN_{\text{A}}T\Rightarrow \frac{1}{2}m_{\text{r}}\langle v^{2}\rangle=\frac{3}{2}RT$$$$$m_{\text{r}}$$ (or $$M$$) is the molar mass (i.e. mass of one mole) of the gas. It is the product of Avogadro's constant and the mass of one particle (i.e. $$m_{r}=N_{A}m$$).

The internal energy of an ideal gas is equivalent to its mean kinetic energy (recall that an ideal gas has negligible intermolecular forces and thus zero potential energy):

$$$U=\frac{3}{2}NkT$$$

The internal energy is directly proportional to the thermodynamic temperature $$T$$.

$$k=$$Boltzmann constant; $$m=$$mass of a particle; $$n=$$number of particles of the gas measured in moles; $$N=$$number of particles, $$R=$$molar gas constant; $$T=$$thermodynamic temperature of the gas; $$U=$$internal energy of the gas.