Question

Supposethat on another planet the atmosphere consists of \(17 \% \mathrm{Kr}, 38 \% \mathrm{CH}_{4}\), and \(45 \% \mathrm{O}_{2}\). What is theaverage 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 is 34.740 g/mol, and the average molar mass at an altitude where all the O2 is photodissociated is 36.988 g/mol.

Step-by-step solution

Find the molar mass of each atmospheric constituent

To find the molar mass of each constituent, we need to know the atomic mass of each element and then multiply the atomic mass by the number of atoms in each molecule. Use the periodic table to find the atomic masses. We'll find the molar mass of Kr, CH4, and O2. For Kr: The molar mass of Kr (Krypton) is 83.798 g/mol. For CH4 (Methane): C (Carbon) has an atomic mass of 12.011 g/mol H (Hydrogen) has an atomic mass of 1.008 g/mol So the molar mass of CH4 is (12.011 + 4 * 1.008) g/mol = 16.043 g/mol For O2 (Oxygen): O (Oxygen) has an atomic mass of 15.999 g/mol So the molar mass of O2 is (2 * 15.999) g/mol = 31.998 g/mol

Calculate the weighted average molar mass at the surface

Now that we have the molar masses for each constituent, we can calculate the weighted average molar mass at the surface using the given percentages. Average Molar Mass (surface) = 0.17 * (Molar Mass of Kr) + 0.38 * (Molar Mass of CH4) + 0.45 * (Molar Mass of O2) = 0.17 * 83.798 g/mol + 0.38 * 16.043 g/mol + 0.45 * 31.998 g/mol = 14.245 g/mol + 6.096 g/mol + 14.399 g/mol = 34.740 g/mol

Calculate the proportions of Kr and CH4 at the higher altitude

At the altitude where all the O2 is photodissociated, there will only be Kr and CH4 present in the atmosphere. To find the new proportions of these gases, we need to normalize the given percentages of Kr and CH4 as if there is no O2 present. Total percentage without O2 = 17% (Kr) + 38% (CH4) = 55% Normalized Kr percentage = 17% / 55% = 0.309 Normalized CH4 percentage = 38% / 55% = 0.691

Calculate the weighted average molar mass at the higher altitude

At the altitude where all the O2 is photodissociated, we can now use the normalized proportions of Kr and CH4 to calculate the weighted average molar mass. Average Molar Mass (high altitude) = 0.309 * (Molar Mass of Kr) + 0.691 * (Molar Mass of CH4) = 0.309 * 83.798 g/mol + 0.691 * 16.043 g/mol = 25.895 g/mol + 11.093 g/mol = 36.988 g/mol The average molar mass at the surface is 34.740 g/mol, and the average molar mass at an altitude where all the O2 is photodissociated is 36.988 g/mol.

Key concepts

Atmospheric Composition

Understanding the composition of a planet's atmosphere is crucial as it influences its chemical and physical characteristics. Each atmosphere is made up of various gases in specific proportions. For example, on Earth, nitrogen and oxygen are dominant. On another planet, like the one mentioned in the problem, the atmosphere consists of 17% krypton (Kr), 38% methane (\(\mathrm{CH}_4\)), and 45% oxygen (\(\mathrm{O}_2\)).
Atmospheric composition is usually described in terms of the volume percentage or molar percentage of each component. This provides insight into the type and behavior of reactions taking place, the climate conditions, and the potential for life. By knowing the atmospheric composition, we can make predictions about how it might change with different conditions, such as altitude where photodissociation of \(\mathrm{O}_2\) occurs.
The molar mass of each constituent must first be identified using the periodic table, and this information provides the foundation for further calculations regarding atmospheric behaviors like density and pressure.

Weighted Average

Calculating the weighted average is fundamental in discerning the molar mass of a planet’s atmosphere. This involves taking the molar mass of each constituent gas and multiplying it by its proportion in the atmosphere.
To find the weighted average molar mass at the surface of this planet, the given percentages are multiplied with their respective molar masses:

• Krypton (Kr): 0.17 * 83.798 g/mol
• Methane (\(\mathrm{CH}_4\)): 0.38 * 16.043 g/mol
• Oxygen (\(\mathrm{O}_2\)): 0.45 * 31.998 g/mol

The result sums to a total molar mass of 34.740 g/mol at the surface.
The concept of weighted average is essential when dealing with mixtures, as it allows one to take into account the relative importance of each component, rather than treating all components equally. At higher altitudes, the percent compositions change with the absence of \(\mathrm{O}_2\), making it necessary to recalculate the weighted average molar mass considering only the remaining gases.

Photodissociation

Photodissociation is a process where molecules are broken down into smaller parts by photons, often from sunlight. It is a significant factor in atmospheric chemistry, particularly at higher altitudes where more UV light is available. For instance, on the planet described in the problem, \(\mathrm{O}_2\) is photodissociated at high altitudes.
When \(\mathrm{O}_2\) gets photodissociated, its molecular bonds are broken, and it no longer contributes to the molar mass calculation at that altitude. This changes the atmospheric composition, leading to a greater relative concentration of the remaining gases. The new molar mass must be recalculated by normalizing the remaining components, krypton and methane.
Understanding photodissociation is crucial for comprehending how solar radiation affects atmospheric layers, leading to phenomena like the formation of an ozone layer or affecting possible habitability by impacting chemical balances.

Periodic Table

The periodic table serves as an indispensable tool in chemistry, providing essential information about the elements that comprise the atmosphere of any planet. Each element on the table is associated with an atomic mass, which is required to compute the molar mass of compounds.
For example, in the problem exercise, the periodic table is used to determine:

• Krypton (Kr) has a molar mass of 83.798 g/mol
• Carbon (C) and hydrogen (H) in methane (\(\mathrm{CH}_4\)) have atomic masses of 12.011 g/mol and 1.008 g/mol respectively, making its molar mass 16.043 g/mol
• Oxygen (\(\mathrm{O}_2\)) has an atomic mass leading to a molar mass of 31.998 g/mol

The periodic table offers a systematic way to obtain the atomic masses required to calculate the molar masses of complex molecules, facilitating the understanding of atmospheric compositions through simple mathematical conversions from atomic to molar scale.