Internal kinetic energy and temperature

König’s decomposition is the basis for understanding what the temperature of a gas is. In an ideal gas the molecules move in a disordered way: the gas’s centre of mass may be at rest (gas in a stationary box) or moving (gas in a moving box), but the individual molecules have much higher speeds, in random directions.

The distinction is crucial: the gas’s overall motion is $E_\text{cin,CM}$, the microscopic agitation is $E_\text{cin,int}$. Only the latter “is” the temperature.

Principle — Internal energy of an ideal gas

The internal kinetic energy of a monatomic ideal gas is tied to temperature: $E_{\text{cin,int}} = \tfrac{3}{2}\,N\,k_B\,T = \tfrac{3}{2}\,n\,R\,T$ where $N$ is the number of particles, $n$ the number of moles, $k_B$ the Boltzmann constant, $R = N_A k_B$ the gas constant.

Hint

Gas in a train travelling at $100$ km/h has a large $E_\text{cin,CM}$, but its temperature depends only on $E_\text{cin,int}$, which stays the same! Heating a gas means increasing the agitation relative to the CM, not making it travel faster. It is the disordered internal motion, not the ordered collective one, that shows up as heat.

The worked exercise Toy gas of two balls shows, with a toy model of just two particles, how you go from $E_\text{cin,int}$ to a “temperature”.

Links

Topics: Centre of mass Concepts: Internal energy · Temperature · Kinetic energy Skills: Micro-macro interpretation

Related exercises: Worked exercise — Toy gas of two balls · Proof of König’s theorem · Internal kinetic energy of two balls