# 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):

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.