Pressure on Venus. At the surface of Venus the average...

Pressure on Venus. At the surface of Venus the average temperature is a balmy $460^{\circ} \mathrm{C}$ 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 $\mathrm{CO}_{2}$ (molar mass $44.0 \mathrm{~g} / \mathrm{mol}$ ), and the temperature remains remarkably constant. Assume that the temperature does not change with altitude. (a) What is the atmospheric pressure $1.00 \mathrm{~km}$ above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres. (b) What is the root-mean-square speed of the $\mathrm{CO}_{2}$ molecules at the surface of Venus and at an altitude of $1.00 \mathrm{~km} ?$

Pressure on Venus. At the surface of Venus the average temperature is a balmy $460^{\circ} \mathrm{C}$ 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 $\mathrm{CO}_{2}$ (molar mass $44.0 \mathrm{~g} / \mathrm{mol}$ ), and the temperature remains remarkably constant. Assume that the temperature does not change with altitude. (a) What is the atmospheric pressure $1.00 \mathrm{~km}$ above the surface of Venus? Express your answer in Venus-atmospheres and earth-atmospheres.
(b) What is the root-mean-square speed of the $\mathrm{CO}_{2}$ molecules at the surface of Venus and at an altitude of $1.00 \mathrm{~km} ?$

Key Concepts

Ideal Gas Law

The Ideal Gas Law relates the pressure, volume, and temperature of a gas to the number of molecules present. This law is fundamental in atmospheric physics as it helps to connect macroscopic quantities like pressure and density with molecular properties, enabling the derivation of expressions for atmospheric pressure profiles and molecular speeds.

Scale Height

Scale height is a characteristic length over which the atmospheric pressure decreases by a factor of e. It combines the temperature, gravitational acceleration, and molecular weight of the gas into a single parameter, and provides an intuitive measure of how quickly the atmosphere thins out with height.

Root-Mean-Square Speed

The root-mean-square speed is a measure of the average speed of particles in a gas, derived from the kinetic theory of gases. It is calculated from the temperature and molar mass of the gas and is essential for understanding molecular motion and the distribution of speeds, which impact various physical properties of the atmosphere.

Hydrostatic Equilibrium

Hydrostatic equilibrium refers to the balance between the upward pressure gradient force and the downward gravitational force in a fluid at rest. In planetary atmospheres, this balance leads to an exponential decrease of pressure with height in an isothermal layer, which is key to understanding variations in atmospheric pressure with altitude.

Isothermal Atmosphere

An isothermal atmosphere is one in which the temperature is assumed to remain constant with altitude. This simplification allows the use of exponential functions to describe how pressure and density diminish with increasing height, making it easier to calculate pressure at different altitudes.

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