Calculate the density of the atmosphere at the surface of Mars (where the pressure is $650$ Pa and the temperature is typically $253$ K, with a CO₂ atmosphere), Venus (with an average temperature of $730$ K and pressure of $92$ atm, with a CO₂ atmosphere), and Saturn's moon Titan (where the pressure is $1.5$ atm and the temperature is $-178$°C, with a N₂ atmosphere).

Verified step by step guidance

1. Identify the ideal gas law equation: \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the number of moles, \( R \) is the ideal gas constant, and \( T \) is temperature in Kelvin.
2. Rearrange the ideal gas law to solve for density \( \rho \): \( \rho = \frac{PM}{RT} \), where \( M \) is the molar mass of the gas.
3. For Mars, use the given pressure \( P = 650 \text{ Pa} \), temperature \( T = 253 \text{ K} \), and the molar mass of CO2 \( M = 44.01 \text{ g/mol} \). Convert the molar mass to kg/mol for consistency with SI units.
4. For Venus, convert the pressure from atm to Pa using \( 1 \text{ atm} = 101325 \text{ Pa} \). Use the given temperature \( T = 730 \text{ K} \) and the molar mass of CO2. Calculate the density using the rearranged ideal gas law.
5. For Titan, convert the pressure from atm to Pa. Convert the temperature from Celsius to Kelvin by adding 273.15. Use the molar mass of N2 \( M = 28.02 \text{ g/mol} \) and calculate the density using the rearranged ideal gas law.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, temperature, and number of moles of a gas. It is expressed as PV = nRT, where P is pressure, V is volume, n is the number of moles, R is the universal gas constant, and T is temperature in Kelvin. This law is crucial for calculating the density of planetary atmospheres by rearranging it to find the density (mass/volume) using known values of pressure and temperature.

Atmospheric Pressure

Atmospheric pressure is the force exerted by the weight of the atmosphere above a given point. It is typically measured in units such as Pascals (Pa) or atmospheres (atm). Understanding atmospheric pressure is essential for calculating the density of a planet's atmosphere, as it directly influences the number of gas molecules present in a given volume, which in turn affects the density.

Molar Mass of Gases

The molar mass of a gas is the mass of one mole of its molecules, typically expressed in grams per mole (g/mol). For calculations involving planetary atmospheres, knowing the molar mass of the primary atmospheric component (e.g., CO2 for Mars and Venus, N2 for Titan) is necessary to determine the density. The molar mass allows conversion between the number of moles and the mass of the gas, which is a key step in applying the Ideal Gas Law to find density.