The Distribution of Molecular Speeds — Question 3

Question 3

A temperature change stretches the distribution while preserving area

The normalized Maxwell speed distribution at temperature $T$ is $f_T(v)$. The same species is heated to $4T$, remaining in classical equilibrium. Its original most probable speed is 300 m/s.

1. Express $f_{4T}(v)$ in terms of $f_T$ and find the new mode and peak height relative to the old peak.

2. If 40.0% of the original population lies between 200 and 500 m/s, identify an interval containing 40.0% after heating and verify the probability transformation.

Question 3 - Solution

A temperature change stretches the distribution while preserving area

(a) Method and calculation

$$f_T(v)\propto T^{-3/2}v^2e^{-mv^2/(2k_BT)}\ \Rightarrow\ f_{4T}(v)=\tfrac12f_T(v/2).$$$$v_{{\mathrm{mp}},4T}=600\,\mathrm{m/s},\quad\max f_{4T}=\tfrac12\max f_T.$$

(b) Interpretation and further calculation

$$P_{4T}(400<v<1000)=\int_{400}^{1000}\tfrac12f_T(v/2)dv.$$$$u=v/2\ \Rightarrow\ P_{4T}=\int_{200}^{500}f_T(u)du=.400.$$Every speed quantile doubles; probability density halves at corresponding rescaled speeds.

Physical check

Heating broadens the normalized curve; it does not increase the total probability.