Pressure, Temperature, and RMS Speed — Question 8

Question 8

Measured pressure and rms speed determine mass density

An isotropic ideal gas has pressure 90.0 kPa and rms speed 600 m/s. Its molecular mass is $5.00\times10^{-26}$ kg. Use $p=\rho v_{\mathrm{rms}}^2/3$ and $k_B=1.381\times10^{-23}$ J/K.

1. Infer mass density, number density and temperature.

2. If the rms-speed reading has hard fractional error at most 1%, with pressure exact, find the exact allowed density ratios relative to the central estimate.

Question 8 - Solution

Measured pressure and rms speed determine mass density

(a) Method and calculation

$$\rho=\frac{3p}{v_{\mathrm{rms}}^2}=\frac{270000}{360000}=.750\,\mathrm{kg/m^3},\quad\nu=\rho/m=1.50\times10^{25}\,\mathrm{m^{-3}}.$$$$T=\frac{mv_{\mathrm{rms}}^2}{3k_B}=\frac{5.00\times10^{-26}(600)^2}{3(1.381\times10^{-23})}\approx434.468\,\mathrm K.$$

(b) Interpretation and further calculation

$$\rho\propto v_{\mathrm{rms}}^{-2}\ \Rightarrow\ \frac{\rho}{\rho_c}\in\left[\frac1{1.01^2},\frac1{.99^2}\right].$$$$\rho/\rho_c\in[.980296,1.020304].$$To first order, the fractional density error has twice the magnitude and opposite sign of the fractional speed error.

Physical check

Number density and mass density differ by molecular mass; interchanging them makes the pressure formula dimensionally wrong.