Answer to Question #124971 in Molecular Physics | Thermodynamics for Gyamfi Maxwell

Question

The density of a gas of relative molecular mass 28 at a certain temperature is 0.90 kgm-3
.
The root mean square speed of the gas molecules at that temperature is 602 ms-1
. Assuming that the rate of a gas is inversely proportional to the square root of it's density calculate the density of the gas standard temperature and pressure of it's root mean square speed is 490m/s.

Answer 1

According to Graham's law, the ratio of root mean square velocities of gases at standard temperature and pressure are inversely proportional to their molecules' masses:

$$\frac{v_1}{v_2}=\sqrt{\frac{m_2}{m_1}}.\\\space\\ m=\Mu n=\rho V=\rho V_mn.$$

Hence:

$$\frac{v_1}{v_2}=\sqrt{\frac{\rho_2 n_2}{\rho_1 n_1}}.$$

For the same amount of substance $n_1=n_2$:

$$\frac{v_1}{v_2}=\sqrt{\frac{\rho_2 }{\rho_1}},\\\space\\ \rho_2=\rho_1\bigg(\frac{v_1}{v_2}\bigg)^2=0.9\bigg(\frac{602}{490}\bigg)^2=1.35\text{ kg/m}^3.$$