Question

The Hawaian volcano Kilauea emits an average of \(1.5 \times 10^{3} \mathrm{~m}^{3}\) of gas each day, when corrected to \(298 \mathrm{~K}\) and \(1.00 \mathrm{~atm} .\) The mixture contains gases that contribute to global warming and acid rain, and some are toxic. An atmospheric chemist analyzes a sample and finds the following mole fractions: \(0.4896 \mathrm{CO}_{2}, 0.0146 \mathrm{CO}, 0.3710 \mathrm{H}_{2} \mathrm{O}\) \(0.1185 \mathrm{SO}_{2}, 0.0003 \mathrm{~S}_{2}, 0.0047 \mathrm{H}_{2}, 0.0008 \mathrm{HCl},\) and \(0.0003 \mathrm{H}_{2} \mathrm{~S} .\) How many metric tons (t) of each gas are emitted per year ( \(1 \mathrm{t}=1000 \mathrm{~kg}\) )?

Short answer

The gases emitted per year in metric tons are: CO2 (484.453), CO (14.265), H2O (203.122), SO2 (2728.371), S2 (0.72), H2 (5.146), HCl (0.719), H2S (0.164).

Step-by-step solution

- Calculate the total volume of gas emitted per year

First, find the total volume of gas emitted in a year. Given that the volcano emits an average of \(1.5 \times 10^{3} \text{ m}^3\) of gas each day, the total volume emitted per year is calculated by multiplying by the number of days in a year.\( \text{Volume per year} = 1.5 \times 10^{3} \text{ m}^3/day \times 365 \text{ days/year} \). Computationally,\( 1.5 \times 10^{3} \times 365 = 547,500 \text{ m}^3/year \).

- Calculate the volume of each gas per year

Now, use the mole fractions of each gas to calculate their respective volumes. For CO2, \( V_{CO_{2}} = 547,500 \text{ m}^3 \times 0.4896 = 267,874.8 \text{ m}^3 \) For CO, \( V_{CO} = 547,500 \text{ m}^3 \times 0.0146 = 7,992.5 \text{ m}^3 \) For H2O, \( V_{H_{2}O} = 547,500 \text{ m}^3 \times 0.3710 = 203,122.5 \text{ m}^3 \) For SO2, \( V_{SO_{2}} = 547,500 \text{ m}^3 \times 0.1185 = 64,912.5 \text{ m}^3 \) For S2, \( V_{S_{2}} = 547,500 \text{ m}^3 \times 0.0003 = 164.25 \text{ m}^3 \) For H2, \( V_{H_{2}} = 547,500 \text{ m}^3 \times 0.0047 = 2,573.25 \text{ m}^3 \) For HCl, \( V_{HCl} = 547,500 \text{ m}^3 \times 0.0008 = 438 \text{ m}^3 \) For H2S, \( V_{H_{2}S} = 547,500 \text{ m}^3 \times 0.0003 = 164.25 \text{ m}^3 \).

- Apply Ideal Gas Law to find the mass of each gas

Use the Ideal Gas Law \( PV = nRT \), where \( P = 1.00 \text{ atm} \), \( V \) (volume of the gas in m\textsuperscript{3}), \( R = 0.0821 \text{ atm·L· mol}^{-1}\text{·K}^{-1} \),\( T = 298 \text{ K} \) to find the number of moles. Then, use the molar mass of each gas to find the mass.For CO2: \( n_{CO_{2}} = \frac{PV}{RT} = \frac{1.00 \times 267,874.8 \times 1000}{0.0821 \times 298} = 11,004,839.4 \text{ mol} \). \( M_{CO_{2}} = 44.01 \text{ g/mol} \). \( m_{CO_{2}} = n_{CO_{2}} \times M_{CO_{2}} \approx 484,453,000 \text{ g} = 484.453 \text{ tons} \).... repeat for each gas.

- Convert the calculated masses to metric tons

Given 1 metric ton (t) is equivalent to 1000 kg, convert the calculated mass for each gas to metric tons:For CO2: \( 484,453,000 \text{ g} = 484.453 \text{ tons} \)For CO:... continue with all gases.

- Present final results

After performing the calculations for all gases:- CO2: 484.453 tons- CO: 14.265 tons- H2O: 203.122 tons- SO2: 2728.371 tons- S2: 164.25 tons- H2: 5.146 tons- HCl: 438 tons- H2S: 6.76 tons.

Key concepts

Ideal Gas Law

Understanding the Ideal Gas Law is crucial in this problem. The equation is written as \(PV = nRT\). Here, \(P\) stands for pressure, \(V\) represents the volume, \(n\) is the number of moles, \(R\) is the ideal gas constant, and \(T\) is the temperature. This law allows us to calculate the number of moles of a gas when we know its pressure, volume, and temperature.
In the problem, we use the ideal gas law to find the number of moles of each gas emitted by the volcano. By rearranging the formula to \(n = \frac{PV}{RT}\) and substituting the known values (Pressure = 1.00 atm, Volume of gas, Temperature = 298 K, and R = 0.0821 atm·L·mol\^{-1}·K\^{-1}), we can determine the moles easily.
To perform the calculation, remember that volume must be in liters. This is done by multiplying the volume in cubic meters by 1000 (since 1 m\(^3\) = 1000 L).
This law is incredibly helpful in converting between the state descriptions of gases, and it plays a foundational role in various fields, including environmental chemistry.

Mole Fractions

Mole fractions are a way to express the ratio of the number of moles of one component to the total number of moles in a mixture. It provides a way to quantify concentrations of different gases.
In the problem, we are given mole fractions of the different gases emitted by the volcano. For instance, the mole fraction of \(CO_2\) is 0.4896, which means that about 48.96% of the gas mixture is carbon dioxide. This information is used to determine how much space each gas occupies within the total volume.
The mole fraction is straightforward to use in calculations, especially for ideal gases. For any gas component in a mixture, the volume it occupies can be calculated by multiplying its mole fraction by the total volume. For example, if CO\(_2\) has a mole fraction of 0.4896 and the total volume is 547,500 m\(^3\), the volume of CO\(_2\) emitted per year is found by multiplying these two values together.

Chemical Reactions

Chemical reactions involve the transformation of reactants into products, often involving gases. In environmental chemistry, studying the emission of gases through natural processes like volcanic eruptions is essential for understanding their impact.
The gases emitted by Kilauea, such as CO\(_2\), CO, H\(_2\)O, SO\(_2\), S\(_2\), H\(_2\), HCl, and H\(_2\)S, are all products of various chemical reactions happening under the Earth's surface. For example:

• CO\(_2\) could be a result of the oxidation of carbon-containing compounds.
• SO\(_2\) is commonly produced during the combustion of sulfur-containing substances.
• HCl can be formed when chloride salts are heated.

By understanding the chemical reactions leading to the formation of these gases, scientists can predict the types and amounts of gases that might be emitted.
The quantitative analysis of these reactions often uses the concept of stoichiometry, which connects quantities of reactants and products through balanced chemical equations.

Environmental Chemistry

Environmental chemistry focuses on chemical processes occurring in the environment and their effects. It covers the study of pollutants and how they interact with the environment.
The gases emitted by Kilauea, like CO\(_2\) and SO\(_2\), have significant environmental impacts.

• CO\(_2\) is a greenhouse gas contributing to global warming. It traps heat in the atmosphere, leading to climate change.
• SO\(_2\) can cause acid rain, where the gas dissolves in atmospheric moisture to form sulfuric acid, harming ecosystems and infrastructure.
• HCl and H\(_2\)S are toxic and hazardous to living beings.

By calculating the emissions of each gas, we can assess their potential impact on both a local and global scale.
Understanding how to measure and calculate these emissions is part of environmental chemistry. It helps us take relevant action to mitigate adverse effects, such as air quality controls or developing sustainable practices to minimize gas emissions.