Chlorofluorocarbons such as \(\mathrm{CCl}_{3} \mathrm{~F}\) \((\mathrm{M}=137.5)\) and \(\mathrm{CCl}_{2} \mathrm{~F}_{2}(\mathrm{M}=121)\) have been linked to ozone depletion in Antarctica. As of 2004 , these gases were found in 275 and 605 parts per trillion \(\left(10^{12}\right)\), by volume. What are the concentrations of these gases under conditions typical of Antarctica stratosphere \((200 \mathrm{~K}\) and \(0.08 \mathrm{~atm}) ?(R=0.081-\mathrm{atm} / \mathrm{K}-\mathrm{mol})\) (a) \(\left[\mathrm{CCl}_{3} \mathrm{~F}\right]=1.375 \times 10^{-12} \mathrm{~mol} \mathrm{1}^{-1}\), \(\left[\mathrm{CCl}_{2} \mathrm{~F}_{2}\right]=3.025 \times 10^{-12} \mathrm{~mol} 1^{-1}\) (b) \(\left[\mathrm{CCl}_{3} \mathrm{~F}\right]=2.75 \times 10^{-14} \mathrm{~mol} 1^{-1},\left[\mathrm{CCl}_{2} \mathrm{~F}_{2}\right]\) \(=6.05 \times 10^{-14} \mathrm{~mol} 1^{-1}\) (c) \(\left[\mathrm{CCl}_{3} \mathrm{~F}\right]=2.75 \times 10^{-10} \mathrm{~mol} 1^{-1},\left[\mathrm{CCl}_{2} \mathrm{~F}_{2}\right]\) \(=6.05 \times 10^{-10} \mathrm{~mol} 1^{-1}\) (d) \(\left[\mathrm{CCl}_{3} \mathrm{~F}\right]=1.375 \times 10^{-13} \mathrm{~mol} \mathrm{H}^{-1}\), \(\left[\mathrm{CCl}_{2} \mathrm{~F}_{2}\right]=3.025 \times 10^{-12} \mathrm{~mol} 1^{-1}\)

Step by step solution

01

Understanding the problem

We are given the partial pressure of the gases in parts per trillion by volume, the molecular masses (M) of the gases, the stratospheric conditions of Antarctica (200 K and 0.08 atm), and the universal gas constant (R). We need to find the concentrations in moles per liter under these conditions.

02

Converting parts per trillion to moles per liter

To convert parts per trillion by volume to concentration, we can use the following relation from the ideal gas law: \(PV = nRT\). Here n is in moles, which is the unit we want for concentration.

03

Calculate the concentration of \(\mathrm{CCl}_{3} \mathrm{F}\)

First, calculate the pressure of \(\mathrm{CCl}_{3} \mathrm{F}\) in atm using the parts per trillion value: \(275 \times 10^{-12}\) atm. Next, find the number of moles per liter \(n/V\) by rearranging the ideal gas law to \(n/V = P/(RT)\), substituting P with the calculated pressure, R with 0.081 atm K^{-1} mol^{-1}, and T with 200 K. This will give us \(\left[\mathrm{CCl}_{3} \mathrm{F}\right]\).

04

Calculate the concentration of \(\mathrm{CCl}_{2} \mathrm{F}_{2}\)

Repeat the same steps as in Step 3, this time with the parts per trillion value for \(\mathrm{CCl}_{2} \mathrm{F}_{2}\) which is \(605 \times 10^{-12}\) atm. After calculations, we get \(\left[\mathrm{CCl}_{2} \mathrm{F}_{2}\right]\).

05

Comparing the results with the given options

Calculate the concentrations and match them with the given options, (a) to (d), to find out which one corresponds to our calculated values.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Parts Per Trillion Conversion

When dealing with chemicals in the atmosphere, concentrations are often so low that they're measured in 'parts per trillion' (ppt). This measurement indicates how many parts of a substance are found in one trillion parts of the mixture.

Understanding ppt is crucial for environmental chemistry, where pollutants may be present at ultra-low concentrations but can still have significant effects. For example, if we say there is a concentration of 275 ppt of a chemical, it means there is 275 parts of the chemical in every one trillion parts of air.

To convert from ppt to a more usable unit such as moles per liter (mol/L), which is necessary to perform calculations about gas behavior under various conditions, we apply the ideal gas law. This involves converting the given volume ratio (ppt) into a pressure, since the ideal gas law relates the pressure of a gas to its number of moles, temperature, and volume (PV = nRT). The conversion is essential in atmospheric chemistry to understand the potential impact of trace gases on processes like ozone depletion.

Ideal Gas Law Application

The ideal gas law, represented by the equation PV = nRT, is a cornerstone of chemical physics, giving us a direct relationship between the pressure (P), volume (V), temperature (T), and number of moles (n) of a gas, with R being the universal gas constant.

For concentration calculations in the atmosphere, we rearrange this equation to solve for the number of moles per liter (n/V), since we are often interested in the molar concentration. Here, P is the partial pressure of the gas, R is the gas constant (0.081 atm K^-1 mol^-1 in the given exercise), and T is the temperature. Assuming we know P and T, we can rearrange the ideal gas law to n/V = P/RT, which gives us the concentration in mol/L after inserting the appropriate values.

This application of the ideal gas law is essential in environmental chemistry for predicting how gases will behave under different atmospheric conditions, such as those found in the stratosphere over Antarctica.

Stratospheric Chemistry

Stratospheric chemistry refers to the chemical processes that occur in the stratosphere, the second major layer of Earth's atmosphere, just above the troposphere. At this altitude, the air is very thin, temperatures are low, and the chemistry is dominated by reactions involving ozone and trace gases, like chlorofluorocarbons (CFCs), which in this case include CCl₃F and CCl₂F₂.

The stratosphere is crucial because it contains the ozone layer, which protects life on Earth by absorbing most of the Sun's harmful ultraviolet radiation. Unfortunately, certain chemicals, specifically CFCs, can lead to ozone layer depletion. Understanding the concentrations of these damaging substances, even when they're present in minute quantities, helps scientists evaluate their impact and informs global policy decisions to protect the ozone layer.

By studying stratospheric chemistry and applying the ideal gas law to calculate the concentration of pollutants like CFCs, we deepen our comprehension of their detrimental effects on the ozone layer, thereby aiding in the development of strategies to mitigate such environmental concerns.