Question

Pressure on Venus. At the surface of Venus the average temperature is a balmy 460°C due to the greenhouse effect (global warming!), the pressure is 92 earth-atmospheres, and the acceleration due to gravity is 0.894 g_earth. The atmosphere is nearly all CO₂ (molar mass 44.0 g/mol), and the temperature remains remarkably constant. Assume that the temperature does not change with altitude. (a) What is the atmospheric pressure 1.00 km above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres. (b) What is the root-mean-square speed of the CO₂ molecules at the surface of Venus and at an altitude of 1.00 km?

Short answer

(a) The atmospheric pressure above the surface of Venus by 1 km in earth-atmospheres is 86.362 atm. Also, the atmospheric pressure above the surface of Venus by 1 km in Venus-atmospheres is 0.939 atm.

(b) The root-mean-square speed at the surface of Venus and at an altitude of 1 km is 644.706 m/s.

Step-by-step solution

Step 1:

Evaluate the atmospheric pressure by the formula

$\mathit{P}\mathbf{=}{\mathit{P}}_{\mathbf{0}}{\mathit{e}}^{\frac{\mathbf{-}\mathbf{M}{\mathbf{a}}_{\mathbf{v}}\mathbf{y}}{\mathbf{R}\mathbf{T}}}$

where M=molar mass of carbon, a_v=acceleration due to the gravity on venus, y=height above the surface, R=gas constant and T= temperature

$$\begin{gathered} p=92atm{e}^{\frac{44\times {10}^{-3}kg/mol\times 0.894\times 9.8m/s^2\times 1\times {10}^{3}m}{8.314J/mol\times \left(460+273.15\right)K}} \\ =86.362atm \end{gathered}$$

The atmospheric pressure above the surface of Venus by 1 km in earth-atmospheres is 86.362 atm.

Also, the atmospheric pressure above the surface of Venus by 1 km in Venus-atmospheres is$\frac{86.392atm}{92atm}=0.939atm$ .

Step 2:

The root mean square speed is given by where R is the molar gas constant,

$$\begin{gathered} u_{rms}=\sqrt{\frac{3RT}{m}} \\ =\sqrt{\frac{3\times 8.314J/mol.K\times \left(460+273.15\right)K}{44\times {10}^{-3}kg/mpl}} \\ =644.706m/s^2 \end{gathered}$$

So, the root-mean-square speed at the surface of Venus and at an altitude of 1 km is 644.706 m/s.