Problem 17 - Physics

Question

On the sunlit surface of Venus, the atmospheric pressure is \(9.0 \times 10^{6} \mathrm{~Pa}\), and the temperature is \(740 \mathrm{~K}\). On the earth's surface the atmospheric pressure is \(1.0 \times 10^{5} \mathrm{~Pa}\), while the surface temperature can reach \(320 \mathrm{~K}\). These data imply that Venus has a "thicker" atmosphere at its surface than does the earth, which means that the number of molecules per unit volume \((N / V)\) is greater on the surface of Venus than on the earth. Find the ratio \((N / V)_{\text {Venus }} /(N / V)_{\text {Earth }}\).

Step-by-Step Solution

Answer

The ratio \( \left( \frac{N}{V} \right)_{Venus} / \left( \frac{N}{V} \right)_{Earth} \) is approximately 51.75.

Step 1: Understand the Ideal Gas Law

The ideal gas law is given by the equation \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant, and \( T \) is the temperature. We can express it in terms of number density \((N/V)\) by rearranging it to \( \frac{N}{V} = \frac{P}{kT} \), where \( k \) is the Boltzmann constant and \( N \) is the number of molecules.

Step 2: Calculate (N/V) for Venus

Using the expression \( \frac{N}{V} = \frac{P}{kT} \), where \( P = 9.0 \times 10^6 \mathrm{~Pa} \) and \( T = 740 \mathrm{~K} \), calculate the molecular density for Venus. Substituting the values, \( \frac{N}{V}_{Venus} = \frac{9.0 \times 10^6}{k \cdot 740} \).

Step 3: Calculate (N/V) for Earth

Using the same formula \( \frac{N}{V} = \frac{P}{kT} \), for Earth with \( P = 1.0 \times 10^5 \mathrm{~Pa} \) and \( T = 320 \mathrm{~K} \), the molecular density for Earth is \( \frac{N}{V}_{Earth} = \frac{1.0 \times 10^5}{k \cdot 320} \).

Step 4: Compute the Ratio (N/V)_Venus / (N/V)_Earth

To find the ratio \( \left( \frac{N}{V} \right)_{Venus} / \left( \frac{N}{V} \right)_{Earth} \), divide the expression for Venus by the expression for Earth: \( \frac{9.0 \times 10^6}{k \cdot 740} \div \frac{1.0 \times 10^5}{k \cdot 320} = \frac{9.0 \times 10^6}{1.0 \times 10^5} \times \frac{320}{740} \). Simplify the ratio to get the final answer.

Key Concepts

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere on a particular surface area. It is measured in Pascals (Pa), a unit of pressure in the International System of Units (SI). Atmospheric pressure can vary depending on the planet, altitude, and weather conditions.
On Venus, the atmospheric pressure on its sunlit surface is notably high at 9.0 million Pa. This indicates a substantial volume of atmospheric gases pressing down on the planet's surface, much more than what we experience here on Earth.
Earth's atmospheric pressure at sea level, which many of us are familiar with, is around 101,325 Pa or 1.0 x 10^5 Pa. This pressure is the result of Earth's atmosphere composed of gases like nitrogen, oxygen, and traces of others.
The difference in atmospheric pressure between the two planets indicates the density and thickness of their atmospheres. A higher pressure on Venus suggests a much denser atmosphere compared to Earth.

Number Density

Number density, represented as \( \frac{N}{V} \), is a measure of the number of particles, such as molecules or atoms, in a given volume. It's important in understanding how concentrated a gas is within an atmosphere.
In the context of Venus and Earth, number density helps explain why Venus's atmosphere is 'thicker'. Since Venus has a higher number of molecules per unit volume, its atmosphere is denser than Earth's.
By using the revised form of the Ideal Gas Law, \( \frac{N}{V} = \frac{P}{kT} \), we can calculate the number density. Here, \( P \) is the pressure, \( k \) is the Boltzmann constant, and \( T \) is the temperature. This approach provides a mathematical representation of how atmospheric conditions influence number density.

Boltzmann Constant

The Boltzmann constant, denoted as \( k \), is a fundamental constant that relates the average kinetic energy of particles in a gas with the temperature of the gas. Its value is approximately \( 1.38 \times 10^{-23} \) J/K.
This constant is essential in the Ideal Gas Law formula \( \frac{N}{V} = \frac{P}{kT} \) to calculate the number density. It acts as a bridge between macroscopic and microscopic physical quantities, helping describe how particles behave at the molecular level.
Without the Boltzmann constant, it would be challenging to link the effects of pressure, temperature, and volume on the microscopic scale of individual molecules.

Temperature Comparison

Temperature plays a crucial role in determining the physical state and behavior of atmospheric gases. It is a measure of the average kinetic energy of the particles in a substance.
Venus has a much higher surface temperature at 740 K compared to Earth's maximum of around 320 K. This elevated temperature on Venus impacts its atmospheric pressure and number density as well.
A higher temperature means that molecules move faster, increasing the likelihood of gaseous expansion and higher pressure. However, despite the high temperature on Venus, its pressure remains immense due to its dense atmosphere.
When comparing the two planets using the formula \( \frac{N}{V} = \frac{P}{kT} \), the differences in temperature provide insight into why the atmospheres behave so differently despite the universal principles governing gases.