Question

The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0⁰C in summer to-100⁰C in winter. Over the course of a Martian year, what are the ranges of (a) the rms speeds of the CO2 molecules and (b) the density (in mol/m3) of the atmosphere?

Short answer

The range of rms speeds of the CO2 molecules is 393m/s to 313m/s

The range of densities of the atmosphere is $0.286\frac{mole}{m^3}$to $0.45\frac{mole}{m^3}$

Step-by-step solution

Data and Formulas

Given data;

Molar mass of CO₂ is $M=44\times {10}^{-3}kg/mole$

Temperature varies from $T=0.0^{0}C=273K$to $T={100}^{0}C=173015K$

Pressure of CO₂is P = 650Pa

Constant R = 8.314 J/mole K

Formulas;

Rms speed

$V_{rms}=\sqrt{\frac{3RT}{M}}$ .......... (1)

Density from Ideal gas formula

$\frac{n}{V}=\frac{P}{RT}$ .......... (2)

Find the rms speeds of the CO2 molecules

For$T=0.0^{0}C=273K$ , put this value in equation (1)

For, put this value in equation (1) and get;

Hence, the range of rms speeds of the CO2 molecules is to

Find the range of densities of the atmosphere

For$T=0.0^{0}C=273K$ , put this value in equation (2)

$$\begin{gathered} \frac{n}{V}=\frac{650Pa}{\left(8.314J/moleK\right)\left(273.15K\right)} \\ =0.286\frac{mole}{m^3} \end{gathered}$$

For$T=-{100}^{0}C=173015K$ , put this value in equation (2) and get;

$\frac{n}{V}=0.45\frac{mole}{m^3}$

Hence, the range of densities of the atmosphere is $0.286\frac{mole}{m^3}$to$0.45\frac{mole}{m^3}$