Pressure, Temperature, and RMS Speed — Question 5

Question 5

Partial pressures add although species have different rms speeds

An ideal mixture contains equal numbers of molecules A and B at one temperature. Their molecular masses satisfy $m_B=4m_A$. Their combined number density is $\nu$, and A has rms speed 800 m/s. Use $p_i=\nu_i m_i\langle v_i^2\rangle/3$ and equipartition $m_i\langle v_i^2\rangle=3k_BT$.

1. Find the rms speed of B and the ratio of partial pressures.

2. Find the number-weighted rms speed of all molecules. Explain why substituting the arithmetic mean molecular mass and that speed into $p=\nu m\langle v^2\rangle/3$ is generally invalid.

Question 5 - Solution

Partial pressures add although species have different rms speeds

(a) Method and calculation

$$m_A(800)^2=4m_Av_{B,\mathrm{rms}}^2\ \Rightarrow\ v_{B,\mathrm{rms}}=400\,\mathrm{m/s}.$$$$p_A/p_B=\nu_A/\nu_B=1.$$

(b) Interpretation and further calculation

$$v_{\mathrm{rms,all}}=\sqrt{\tfrac12(800^2+400^2)}=\sqrt{400000}=632.456\,\mathrm{m/s}.$$$$\langle mv^2\rangle=m_A(640000),\quad\langle m\rangle\langle v^2\rangle=(2.5m_A)(400000)=1000000m_A.$$Mass and speed squared are correlated across species; the product of separate averages overestimates the correct moment by a factor $25/16$.

Physical check

At equilibrium each molecule contributes the same mean translational energy, but not the same mean square speed.