Question

Natural gas consists of about \(90 \%\) methane, \(\mathrm{CH}_{4}\) Assume that the solubility of natural gas at \(20^{\circ} \mathrm{C}\) and 1 atm gas pressure is about the same as that of \(\mathrm{CH}_{4}\) \(0.02 \mathrm{g} / \mathrm{kg}\) water. If a sample of natural gas under a pressure of 20 atm is kept in contact with \(1.00 \times 10^{3} \mathrm{kg}\) of water, what mass of natural gas will dissolve?

Short answer

The mass of natural gas that will dissolve in the water is 400 g.

Step-by-step solution

Understand Henry's Law

Henry's law states that 'At a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.' In this case, we are given the solubility at 1 atm (0.02g in 1kg of water) and we are asked to find the solubility at 20 atm.

Apply the Law

Since the solubility of the gas is directly proportional to its partial pressure, the solubility of gas at 20 atm would be 20 times the solubility at 1 atm. This is because the pressure has increased by a factor of 20. So, the solubility of gas at 20 atm is \(20 \times 0.02 \, g/kg = 0.4 \, g/kg\). This means 0.4 g of gas will dissolve in every kg of water at 20 atm.

Calculate the total mass of dissolved gas

To find the total mass of the gas that will dissolve, multiply the solubility of the gas at 20 atm by the total mass of the water. In this case, the water mass is \(1.00 \times 10^{3} \, kg\), so \(0.4 \, g/kg \times 1.00 \times 10^{3} \, kg = 400 \, g\). Therefore, 400 g of natural gas will dissolve in the water.

Key concepts

Natural Gas Solubility

When we're talking about natural gas, much of it is made up of methane, which has a noted solubility in water. Solubility refers to the maximum amount of a substance that can be dissolved in a solvent at a given temperature and pressure. For natural gas, the solubility is often modeled using its predominant component, methane. At 20 degrees Celsius and under a pressure of 1 atm, methane's solubility is 0.02 grams per kilogram of water. This means that at standard pressure, only a small amount of methane can dissolve in water.
Understanding gas solubility is crucial, especially in industries where the dissolution of gases impacts processes, like in natural gas extraction and processing. By learning about gas solubility, you gain insights into how gases interact with water and other liquids under varying conditions.

Partial Pressure

In the context of gases, partial pressure is a fundamental concept that helps us determine how much gas can dissolve in a liquid. According to Dalton's Law, each gas in a mixture exerts pressure independently of others. The partial pressure of a gas like methane in a mixture is the pressure that gas would have if it alone occupied the entire volume.
For natural gas solubility, Henry's Law incorporates this idea by stating that a gas's solubility in a liquid is directly proportional to the gas's partial pressure in contact with the liquid. Thus, if the pressure of methane, or any gas, above a solvent increases, more of it will dissolve in the solvent. In simple words, the more the gas pushes against the liquid, the more it goes into solution. So, understanding partial pressure helps us manipulate and predict the amount of dissolved gas.

Gas Solubility Calculation

Calculating gas solubility is a straightforward application of Henry’s Law. In the given problem, with an initial solubility known at 1 atm, we can deduce the solubility at 20 atm by multiplying the known solubility by the change in pressure. Since the pressure increased twentyfold, the solubility increases in direct proportion, resulting in a solubility of 0.4 g of gas per kg of water.
To find out how much total gas dissolves in a given volume of water, you multiply the solubility by the mass of the water. With 1,000 kg of water at a solubility of 0.4 g/kg, the total mass of dissolved gas equals 400 grams. These calculations are essential in predicting and managing natural gas behaviors and interactions, especially under varying pressures.