Answer
(a)
Calculate the density of the Mars at Martian atmosphere when $T=-50^{\circ} \mathrm{C}$ :
$$
\rho_{\text {mars }}=\frac{p_{\text {mars }}}{R T_{\text {mars }}}
$$
Here, the pressure at Mars is $p_{\max }$, the gas constant is $R$, the temperature at the surface of Mars is $T_{\text {mars }}$
As Gas constant value is equivalent to carbon dioxide, $R=188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$
Substitute, $900 \frac{\mathrm{N}}{\mathrm{m}^{2}}$ for $p_{\text {mars }}, 188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ for $R, 223 \mathrm{~K}$ for $T$
$$
\begin{aligned}
\rho_{\operatorname{mars}} &=\frac{900 \frac{\mathrm{N}}{\mathrm{m}^{2}}}{\left(188.9 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\right)(223 \mathrm{~K})} \\
&=0.0214 \frac{\mathrm{kg}}{\mathrm{m}^{3}}
\end{aligned}
$$
Calculate the density of the earth:
$$
\rho_{\text {earlh }}=\frac{p_{\text {earth }}}{R T_{\text {carth }}}
$$
Here, the pressure at earth is $p_{\text {earth }}$, the gas constant is $R$, the temperature at the surface of Mars is $T_{\text {earth }}$
Gas constant of air $R=286.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$
Substitute, $101.6 \times 10^{3} \frac{\mathrm{N}}{\mathrm{m}^{2}}$ for $p_{\text {mars }}, 188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ for $R, 223 \mathrm{~K}$ for $T$
$$
\begin{aligned}
\rho_{\operatorname{mars}} &=\frac{101.6 \times 10^{3} \frac{\mathrm{N}}{\mathrm{m}^{2}}}{\left(286.9 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\right)(291 \mathrm{~K})} \\
&=1.22 \frac{\mathrm{kg}}{\mathrm{m}^{3}}
\end{aligned}
$$
Work Step by Step
(a)
Calculate the density of the Mars at Martian atmosphere when $T=-50^{\circ} \mathrm{C}$ :
$$
\rho_{\text {mars }}=\frac{p_{\text {mars }}}{R T_{\text {mars }}}
$$
Here, the pressure at Mars is $p_{\max }$, the gas constant is $R$, the temperature at the surface of Mars is $T_{\text {mars }}$
As Gas constant value is equivalent to carbon dioxide, $R=188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$
Substitute, $900 \frac{\mathrm{N}}{\mathrm{m}^{2}}$ for $p_{\text {mars }}, 188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ for $R, 223 \mathrm{~K}$ for $T$
$$
\begin{aligned}
\rho_{\operatorname{mars}} &=\frac{900 \frac{\mathrm{N}}{\mathrm{m}^{2}}}{\left(188.9 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\right)(223 \mathrm{~K})} \\
&=0.0214 \frac{\mathrm{kg}}{\mathrm{m}^{3}}
\end{aligned}
$$
Calculate the density of the earth:
$$
\rho_{\text {earlh }}=\frac{p_{\text {earth }}}{R T_{\text {carth }}}
$$
Here, the pressure at earth is $p_{\text {earth }}$, the gas constant is $R$, the temperature at the surface of Mars is $T_{\text {earth }}$
Gas constant of air $R=286.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$
Substitute, $101.6 \times 10^{3} \frac{\mathrm{N}}{\mathrm{m}^{2}}$ for $p_{\text {mars }}, 188.9 \mathrm{~J} / \mathrm{kg} \cdot \mathrm{K}$ for $R, 223 \mathrm{~K}$ for $T$
$$
\begin{aligned}
\rho_{\operatorname{mars}} &=\frac{101.6 \times 10^{3} \frac{\mathrm{N}}{\mathrm{m}^{2}}}{\left(286.9 \frac{\mathrm{J}}{\mathrm{kg} \cdot \mathrm{K}}\right)(291 \mathrm{~K})} \\
&=1.22 \frac{\mathrm{kg}}{\mathrm{m}^{3}}
\end{aligned}
$$
