Question

(a) 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 \(CO_2\) atmosphere), Venus (with an average temperature of 730 \(K\) and pressure of 92 atm, with a \(CO_2\) atmosphere), and Saturn's moon Titan (where the pressure is 1.5 atm and the temperature is -178\(^\circ\)C, with a \(N_2\) atmosphere). (b) Compare each of these densities with that of the earth's atmosphere, which is 1.20 kg/m\(^3\). Consult Appendix D to determine molar masses.

Short answer

Earth's atmosphere is denser than Mars, but less dense than Venus and Titan.

Step-by-step solution

Understand the Ideal Gas Law

The Ideal Gas Law equation is \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the universal gas constant (8.314 J/mol·K), and \( T \) is the temperature in Kelvin. We need to find the density \( \rho \), which is mass per unit volume \( m/V \). To find \( \rho \), we rearrange it in terms of \( P \) and \( T \): \[ \rho = \frac{PM}{RT} \] where \( M \) is the molar mass of the gas.

Determine Molar Masses

For the atmospheres in question:- Mars and Venus have mainly \( CO_2 \), with a molar mass of about 44.01 g/mol.- Titan has primarily \( N_2 \), with a molar mass of about 28.02 g/mol.

Mars Density Calculation

Given: \( P = 650 \; Pa \), \( T = 253 \; K \), \( M = 44.01 \; g/mol = 0.04401 \; kg/mol \).Substitute these into the density formula:\[ \rho_{Mars} = \frac{650 \times 0.04401}{8.314 \times 253} = 0.020 \; kg/m^3 \]

Venus Density Calculation

Convert the pressure from atm to Pa: \( 92 \; atm = 92 \times 101325 \; Pa = 9316300 \; Pa \).Given: \( T = 730 \; K \), \( M = 44.01 \; g/mol = 0.04401 \; kg/mol \).Substitute into the density formula:\[ \rho_{Venus} = \frac{9316300 \times 0.04401}{8.314 \times 730} = 67.97 \; kg/m^3 \]

Titan Density Calculation

Convert the temperature to Kelvin: \( T = -178^\circ C = 273.15 - 178 = 95.15 \; K \).Convert the pressure from atm to Pa: \( 1.5 \; atm = 1.5 \times 101325 \; Pa = 151987.5 \; Pa \).Given: \( M = 28.02 \; g/mol = 0.02802 \; kg/mol \).Substitute into the density formula:\[ \rho_{Titan} = \frac{151987.5 \times 0.02802}{8.314 \times 95.15} = 5.38 \; kg/m^3 \]

Compare with Earth's Density

The calculated densities are: - Mars: 0.020 kg/m^3 - Venus: 67.97 kg/m^3 - Titan: 5.38 kg/m^3 Earth's atmospheric density is 1.20 kg/m^3. Thus, Venus has a much denser atmosphere than Earth's, Titan is denser but less so compared to Earth, and Mars has a much thinner atmosphere.

Key concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental principle in chemistry and physics for understanding how gases behave. It relates several properties of gases in a simple equation: \( PV = nRT \).

• Pressure (P) is the force exerted by the gas particles on the walls of its container, measured in Pascals (Pa) or atmospheres (atm).
• Volume (V) is the amount of space that the gas occupies, usually measured in cubic meters (m³) or liters (L).
• Number of Moles (n) indicates the quantity of gas particles.
• Universal Gas Constant (R) has a value of 8.314 J/(mol·K) and connects all these variables.
• Temperature (T) is expressed in Kelvin (K) and affects how fast the gas particles move.

To calculate the density of a gas using the Ideal Gas Law, we rearrange it to \( \rho = \frac{PM}{RT} \), where \( \rho \) is the density, \( M \) is the molar mass, and the other symbols have their usual meanings. It's essential for comparing different planetary atmospheres.

Molar Mass

Molar mass is a critical concept when dealing with gases, as it tells us the mass of one mole of a substance, typically expressed in grams per mole (g/mol). Each molecule consists of atoms with well-defined masses, leading to a molar mass specific to that compound.
For example:

• Carbon dioxide \( (CO_2) \) has a molar mass of approximately 44.01 g/mol. It's the main component of both Mars and Venus atmospheres.
• Nitrogen \( (N_2) \), which primarily makes up Titan's atmosphere, has a molar mass of about 28.02 g/mol.

Understanding molar mass is crucial because it determines how much one mole of a gas weighs and influences the calculation of density. Heavier gases, like \( CO_2 \), result in higher density under the same conditions compared to lighter gases like \( N_2 \). Molar mass must be converted to kilograms per mole when plugging into the Ideal Gas Law formula for density calculation.

Planetary Atmospheres

Planetary atmospheres vary significantly in composition, pressure, and temperature, affecting their overall density. Each planet or moon has a unique atmosphere:

• Mars: Dominated by \( CO_2 \), it has a high-altitude, thin atmosphere, with low pressure at 650 Pa and a temperature of around 253 K. This leads to a very low density of around 0.020 kg/m³.
• Venus: Has an incredibly dense atmosphere composed mainly of \( CO_2 \). The pressure is immense at 92 atm, and the average temperature is a scorching 730 K, resulting in a much higher density of approximately 67.97 kg/m³.
• Titan (moon of Saturn): The atmosphere is primarily \( N_2 \), with a pressure of 1.5 atm and a surface temperature of -178 °C (95.15 K), leading to a density of about 5.38 kg/m³.

These densities inform us about how different these celestial bodies are from Earth, whose atmosphere has a density of 1.20 kg/m³, a mix of oxygen and nitrogen at much lower pressures and higher temperatures compared to Venus.

Pressure Conversion

Pressure conversion is a necessary step when comparing atmospheric data, as measurements are often given in different units.
The common units of pressure include:

• Pascal (Pa), the SI unit of pressure, commonly used in scientific calculations.
• Atmosphere (atm), a unit based on the average atmospheric pressure at sea level on Earth.

To convert pressure from atmospheres to Pascals, use the conversion factor:\[ 1 \text{ atm} = 101325 \text{ Pa} \]For example, Venus's atmospheric pressure of 92 atm converts to 9316300 Pa by multiplying 92 by 101325. This conversion provides accuracy in calculations where pressure directly affects the derived quantities such as density. Similarly, Titan's 1.5 atm becomes 151987.5 Pa. These conversions ensure consistency when applying the Ideal Gas Law, allowing for accurate density calculations.