Problem 62 - Chemistry The Central Science In Si Units

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?

Step-by-Step Solution

Answer

The average molar mass at the surface is 30.78 g/mol; at altitude, it is 29.56 g/mol.

Step 1: Understanding the Components at Surface

The atmosphere on the planet consists of Krypton (Kr), Methane (CH\(_4\)), and Oxygen (O\(_2\)). By percentage, the composition is 10% Kr, 40% CH\(_4\), and 50% O\(_2\).

Step 2: Determine the Molar Mass of Each Component

The molar mass of Kr is approximately 83.8 g/mol, CH\(_4\) is approximately 16.0 g/mol, and O\(_2\) is 32.0 g/mol.

Step 3: Calculate Average Molar Mass at the Surface

The average molar mass at the surface is calculated using the formula: \[ \text{Average Molar Mass} = (0.1 \times 83.8) + (0.4 \times 16.0) + (0.5 \times 32.0) \] After calculation: \[ = 8.38 + 6.4 + 16.0 = 30.78 \text{ g/mol} \]

Step 4: Atmospheric Composition at Altitude

At the altitude where all the O\(_2\) is photodissociated, we are left with only Kr and CH\(_4\) in their previous proportions of 20% Kr and 80% CH\(_4\) (as these take over the percentage previously held by O\(_2\)).

Step 5: Molar Mass Calculation at Altitude

The average molar mass at this altitude can be calculated by: \[ \text{Average Molar Mass} = (0.2 \times 83.8) + (0.8 \times 16.0) \] After calculation: \[ = 16.76 + 12.8 = 29.56 \text{ g/mol} \]

Step 6: Conclusion: Results Interpretation

The average molar mass at the surface is 30.78 g/mol, while at the altitude where all O\(_2\) is photodissociated, it is 29.56 g/mol.

Key Concepts

Atmospheric Composition

The atmosphere of a planet is made up of various gases. The composition and proportions of these gases influence the properties of the atmosphere like its molar mass. For example, the planet mentioned in the exercise has an atmosphere composed of

• 10% Krypton (Kr)
• 40% Methane (CH₄)
• 50% Oxygen (O₂)

Understanding atmospheric composition is crucial as it affects everything from the average molar mass to the chemical and physical behavior of the atmosphere. Each gas in an atmosphere contributes to its collective properties depending on the volume percentage and individual gas characteristics. Changes in atmospheric composition can result from natural processes or external influences, significantly affecting conditions on the surface and at various altitudes.

Molar Mass Calculation

Molar mass calculation is the process of determining the mass of one mole of a substance. It's an essential concept when studying chemistry and helps understand the properties of gases in an atmosphere. To calculate the average molar mass of a mixture, the formula used is:\[ \text{Average Molar Mass} = \sum (\text{Fraction of Gas} \times \text{Molar Mass of Gas}) \]In the example provided, you learn how this calculation works with a mixture of Kr, CH₄, and O₂. At the planet's surface, the average molar mass is calculated by:

• Multiplying the percentage of each gas by its molar mass
• Summing up these values

This yields a molar mass of 30.78 g/mol. This value changes with atmospheric conditions, demonstrating the importance of understanding molar mass in various contexts.

Photodissociation

Photodissociation refers to the process in which molecules are broken down into smaller components due to absorption of light. In atmospheres, this process commonly affects gases like O₂. The exercise illustrates how O₂ can be photodissociated at certain altitudes, altering the atmospheric composition. As the O₂ is dissociated, only Kr and CH₄ are left, taking up its previous percentages. Photodissociation is influenced by wavelengths of light, as certain components absorb specific wavelengths, leading to more or less photodissociation. The change from 50% O₂ to none affects the molar mass which decreases to 29.56 g/mol at that altitude. Comprehending photodissociation helps in understanding the dynamic nature of atmospheres and how solar radiation impacts planetary environments.