Maxwell–Boltzmann Speeds, Energy Distributions, and Applications

All flashcards

What does the Maxwell–Boltzmann speed distribution describe?

The statistical spread of molecular speeds in a gas at a specified temperature.

How do most probable, average, and rms speeds differ?

They describe the distribution’s peak, the mean speed, and the square root of the mean squared speed, respectively.

How are speed and kinetic-energy probability densities related?

They assign equal probability to corresponding intervals: $f_K(K)\,dK=f(v)\,dv$.

Why does evaporative cooling lower a liquid’s temperature?

Escaping molecules tend to have higher energy, lowering the average energy of those left behind.

Why are light gases more likely to escape an atmosphere?

At the same temperature, lighter molecules have higher characteristic speeds and are more likely to reach escape speeds.

Quiz questions

1. For a Maxwell–Boltzmann gas, how are the most probable speed $v_p$, average speed $\bar v$, and rms speed $v_{\mathrm{rms}}$ ordered?Answer: $v_p<\bar v<v_{\mathrm{rms}}$
2. For a molecule with kinetic energy $K=\tfrac12 mv^2$, how are the speed and energy probability densities related?Answer: $f_K(K)\,dK=f(v)\,dv$
3. Why are hydrogen and helium molecules more likely than nitrogen molecules to reach Earth’s atmospheric escape speeds at the same temperature?Answer: Their lower molecular masses give them higher characteristic speeds.
4. Why does evaporation tend to cool the liquid that remains?Answer: Higher-energy molecules escape, reducing the average energy of those left behind.
5. In a continuous molecular-speed distribution, what is the probability of finding a molecule at one exact specified speed?Answer: Zero; probabilities are assigned to speed intervals.