Question

Suppose that 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 of the planet is \(28.67\: g/mol\), and the average molar mass at an altitude where all \(O_2\) is photodissociated is \(36.91\: g/mol\).

Step-by-step solution

Find the molar masses of given gases

To find the molar masses of the given gases, we will make use of the periodic table. The molar mass of each gas can be calculated by adding the molar mass of each element multiplied by its quantity in the molecule. Molar mass of \(Kr\): \(83.80\: g/mol\) Molar mass of \(CH_4\): \(12.01\: g/mol (C) + 4 \times 1.01\: g/mol (H) = 16.05\: g/mol\) Molar mass of \(O_2\): \(2 \times 16.00\: g/mol (O) = 32\: g/mol\)

Calculate the average molar mass at the surface

Now, use the given percentages and molar masses of the gases to calculate the average molar mass at the surface: Average molar mass at the surface = \(\frac{17\% \times 83.80\: g/mol (Kr) + 38\% \times 16.05\: g/mol (CH_4) + 45\% \times 32\: g/mol (O_2)}{100}\) Average molar mass at the surface = \(\frac{0.17 \times 83.80 + 0.38 \times 16.05 + 0.45 \times 32}{1}\) Average molar mass at the surface = \(28.67\: g/mol\)

Calculate the average molar mass at an altitude where all the \(O_2\) is photodissociated

At an altitude where all the \(O_2\) is photodissociated, the percentage of \(O_2\) in the atmosphere becomes 0%. The percentages of the other gases remain the same. Therefore, we will recalculate the average molar mass without the contribution of \(O_2\): Average molar mass at altitude = \(\frac{17\% \times 83.80\: g/mol (Kr) + 38\% \times 16.05\: g/mol (CH_4)}{17+38}\) Average molar mass at altitude = \(\frac{0.17 \times 83.80 + 0.38 \times 16.05}{0.55}\) Average molar mass at altitude = \(36.91\: g/mol\) To conclude, the average molar mass at the surface of the planet is \(28.67\: g/mol\), and the average molar mass at an altitude where all \(O_2\) is photodissociated is \(36.91\: g/mol\).

Key concepts

Atmospheric Composition

The atmosphere of any planet is a mixture of different gases. These gases are found at various percentages that define the atmospheric composition. Let's explore what's contained in the atmosphere of a fictional planet we are discussing. It consists of three primary gases: Krypton (Kr), Methane (CH extsubscript{4}), and Oxygen (O extsubscript{2}). Each of these gases has its own unique molar mass, a property that affects how we calculate the average molar mass of the entire atmosphere.

To calculate the average molar mass, the molar masses of individual gases and their percentages in the atmosphere are required. On this planet, Krypton makes up 17% of the atmosphere, Methane constitutes 38%, and Oxygen is present at 45%. Despite their percentages, at varying altitudes, the composition can change based on external processes such as photodissociation, which we will address next.

Photodissociation

Photodissociation is an essential chemical process in atmospheric science. It involves the breaking down of molecules into their constituent atoms due to the absorption of light, particularly ultraviolet radiation. In the context of our fictional planet, photodissociation predominantly affects the Oxygen (O extsubscript{2}) molecules in the atmosphere at higher altitudes.

When atmospheric Oxygen is exposed to high-energy ultraviolet light at specific altitudes, the bonds between oxygen atoms break, resulting in separate Oxygen atoms. This reduces the concentration of O extsubscript{2}, consequently altering the atmospheric composition and influencing the molar mass. Since the O extsubscript{2} percentage drops to zero, we can calculate how this change affects the overall average molar mass of the atmosphere. Photodissociation is a crucial process as it can significantly impact atmospheric properties and behavior.

Periodic Table

The periodic table is an invaluable tool for calculating molar masses. It lists all known elements, providing essential data such as atomic masses, which are crucial for our calculations. This universal chart allows us to determine the molar masses of compounds, like those found in the planetary atmosphere.

For instance, using the periodic table, we find the molar mass of Krypton (Kr) to be 83.80 g/mol. Methane (CH extsubscript{4}) is calculated as 16.05 g/mol by adding the carbon atom's mass (12.01 g/mol) and four hydrogen atoms' mass (4 x 1.01 g/mol). Oxygen (O extsubscript{2}), consisting of two oxygen atoms, has a molar mass of 32 g/mol. By using these molar masses, we calculate the average molar mass of the atmosphere.

Knowing these values is essential to solving problems involving atmospheric composition and any changes that may occur, such as during photodissociation.

Gas Mixture

A gas mixture consists of multiple gas components that experience some interaction or co-existence in a shared space, such as the atmosphere of the planet we're examining. Each gas contributes to the properties of the entire mixture, like density and molar mass.

To find the average molar mass of our planetary atmosphere, we use the percentage contribution of each gas in combination with their molar masses. Initially, we account for all components: Krypton, Methane, and Oxygen. However, the behavior of the gas mixture can change notably with altitude due to processes like photodissociation. This is because the gas concentrations and percentages shift as atmospheric conditions vary.

By understanding gas mixtures, we comprehend how individual gas properties affect the whole, impacting calculations such as the average molar mass and consequently the characteristics of the atmosphere.