Question

Suppose that on another planet the atmosphere consists of \(10 \% \mathrm{Kr}, 40 \% \mathrm{CH}_{4},\) and \(50 \% \mathrm{O}_{2} .\) What is the average molar mass at the surface? What is the average molar mass at an altitude at which all the \(\mathrm{O}_{2}\) is photodissociated?

Short answer

The average molar mass at the surface of the planet is \(30.796 g/mol\), and the average molar mass at the altitude where all the \(O_{2}\) is photodissociated is \(14.796 g/mol\).

Step-by-step solution

Calculate the surface average molar mass

To calculate the average molar mass at the surface, first find the molar mass of each gas Kr, CH₄, and O₂. Then, multiply these values by their respective volume percentages and sum the results to obtain the average molar mass. Molar mass of Kr = 83.8 g/mol Molar mass of CH₄ = 16.04 g/mol Molar mass of O₂ = 32.00 g/mol Surface average molar mass = (0.10)(83.8 g/mol) + (0.40)(16.04 g/mol) + (0.50)(32.00 g/mol) = 8.38 g/mol + 6.416 g/mol + 16.00 g/mol

Add the individual contributions to find the average molar mass at the surface

Sum the contributions from each gas at the surface to find the final average molar mass: Surface average molar mass = 8.38 g/mol + 6.416 g/mol + 16.00 g/mol = 30.796 g/mol

Calculate the average molar mass at the altitude where O₂ is photodissociated

Since all O₂ is photodissociated, we only need to consider the contributions from Kr and CH₄. Using the same method as above, calculate the average molar mass without O₂: No O₂ average molar mass = (0.1)(83.8 g/mol) + (0.4)(16.04 g/mol) = 8.38 g/mol + 6.416 g/mol

Add the individual contributions to find the average molar mass at the altitude without O₂

Sum the contributions from Kr and CH₄ to find the final average molar mass: No O₂ average molar mass = 8.38 g/mol + 6.416 g/mol = 14.796 g/mol The average molar mass at the surface of the planet is 30.796 g/mol, and the average molar mass at the altitude where all the O₂ is photodissociated is 14.796 g/mol.

Key concepts

Understanding Atmospheric Composition

The atmospheric composition of any planet refers to the different gases that make up its atmosphere. On Earth, for example, we have a mix of nitrogen, oxygen, carbon dioxide, and other gases. Understanding the composition is crucial, as it affects the planet's climate, weather, and even the potential for life.

On the hypothetical planet in our exercise, the atmosphere is made up of 10% krypton (Kr), 40% methane (CH₄), and 50% oxygen (O₂). The percentage indicates the proportion each gas contributes to the total volume of the atmosphere.

When analyzing a planet's atmosphere, consider:

• The type of gases present and their chemical properties
• The relative percentages of each gas
• How these gases affect processes such as temperature regulation and radiation shielding

Mastering Molar Mass Calculation

Calculating the average molar mass of a gas mixture involves knowing the molar mass of each component and their respective volume percentages. Molar mass is the mass of a given substance (g) divided by the amount of substance (mol), and it’s typically expressed in g/mol.

To find the average molar mass of an atmosphere:

• Identify the molar mass of each component gas. For example, krypton's molar mass is 83.8 g/mol, methane's is 16.04 g/mol, and oxygen's is 32.00 g/mol.
• Multiply the molar mass of each gas by its volume fraction in the atmosphere. This gives the contribution of each gas to the average molar mass.
• Add these contributions together to get the total average molar mass.

For the surface of the given planet, the calculation yields an average molar mass of 30.796 g/mol. This value changes with the composition of the atmosphere, particularly when some gases like O₂ undergo photodissociation, as we shall see next.

Exploring Photodissociation

Photodissociation is a process where molecules are broken down into smaller units due to the absorption of photons, particularly from solar radiation. On Earth, photodissociation is a key player in the chemical changes occurring in the stratosphere, affecting the ozone layer.

In the context of the hypothetical planet, all oxygen ( O₂) molecules are photodissociated at a certain altitude. This leads to significant changes in the atmospheric composition and hence its average molar mass. Without oxygen, only krypton and methane contribute to the molar mass.

When calculating the new average molar mass:

• O₂ is removed from the calculation, so only the remaining gases are considered.
• The contribution of krypton and methane is recalculated to find the new molar mass.

The average molar mass plummets to 14.796 g/mol, highlighting how photodissociation can drastically alter atmospheric properties and potentially local climate conditions.