Martian Climate. The atmosphere of Mars is mostly CO2 (molar mass 44.0 g/mol) under a pressure of 650 Pa, which we shall assume remains constant. In many places the temperature varies from 0.0°C in summer to -100°C in winter. Over the course of a Martian year, what are the ranges of (b) the density (in mol/m^3) of the atmosphere?

Verified step by step guidance

1. First, understand that the ideal gas law is applicable here, which is given by the equation: <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>P</mi><mi>V</mi>=<mi>n</mi><mi>R</mi><mi>T</mi></math>, where <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>P</mi></math> is the pressure, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>V</mi></math> is the volume, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>n</mi></math> is the number of moles, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi></math> is the ideal gas constant, and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>T</mi></math> is the temperature in Kelvin.
2. Convert the given temperatures from Celsius to Kelvin. The conversion formula is <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>T</mi>_k = <mi>T</mi>_c + 273.15</mi></math>. So, for summer, <math xmlns='http://www.w3.org/1998/Math/MathML'><mn>0.0</mn>°<mi>C</mi></math> becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mn>273.15</mn> <mi>K</mi></math>, and for winter, <math xmlns='http://www.w3.org/1998/Math/MathML'><mn>-100</mn>°<mi>C</mi></math> becomes <math xmlns='http://www.w3.org/1998/Math/MathML'><mn>173.15</mn> <mi>K</mi></math>.
3. Rearrange the ideal gas law to solve for the number of moles per unit volume (density in mol/m³): <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>n</mi>/<mi>V</mi> = <mi>P</mi>/(<mi>R</mi><mi>T</mi>)</math>. Here, <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>P</mi> = <mn>650</mn> <mi>Pa</mi></math> and <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>R</mi> = <mn>8.314</mn> <mi>J</mi>/(<mi>mol</mi> <mi>K</mi>)</math>.
4. Calculate the density for the summer temperature: Substitute <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>T</mi> = <mn>273.15</mn> <mi>K</mi></math> into the rearranged equation to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>n</mi>/<mi>V</mi></math> for summer.
5. Calculate the density for the winter temperature: Substitute <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>T</mi> = <mn>173.15</mn> <mi>K</mi></math> into the rearranged equation to find <math xmlns='http://www.w3.org/1998/Math/MathML'><mi>n</mi>/<mi>V</mi></math> for winter. This will give you the range of densities over the Martian year.

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation in physics that relates the pressure, volume, temperature, and amount of gas. It is expressed as 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. This law is crucial for calculating the density of gases under varying conditions.

Temperature Conversion

Temperature conversion is essential when working with the Ideal Gas Law, as it requires temperature in Kelvin. To convert from Celsius to Kelvin, add 273.15 to the Celsius temperature. For example, 0°C is 273.15 K, and -100°C is 173.15 K. Accurate conversion ensures correct calculations in thermodynamic equations.

Density of a Gas

The density of a gas is defined as its mass per unit volume, often expressed in mol/m^3 for gases. Using the Ideal Gas Law, density can be calculated as n/V = P/RT, where n/V is the molar density. Understanding how pressure and temperature affect density is key to solving problems involving atmospheric conditions on planets like Mars.