Physics

The key insight is that temperature in a gas is fundamentally a measure of random molecular motion. For a monatomic ideal gas, all the thermal energy goes into translational kinetic energy (no rotation or vibration), so there's a direct proportional relationship between $T$ and $K_{\text{avg}}$. The equipartition theorem tells us that each translational degree of freedom (motion in $x$, $y$, and $z$ directions) contributes equally to the average energy, giving you a simple factor involving Boltzmann's constant $k_B$.

The key insight is that temperature in a gas is fundamentally a measure of random molecular motion. For a monatomic ideal gas, all the thermal energy goes into translational kinetic energy (no rotation or vibration), so there's a direct proportional relationship between $T$ and $K_{\text{avg}}$. The equipartition theorem tells us that each translational degree of freedom (motion in $x$, $y$, and $z$ directions) contributes equally to the average energy, giving you a simple factor involving Boltzmann's constant $k_B$.