Question

The Henry's law constant for helium gas in water at \(30^{\circ} \mathrm{C}\) is \(3.7 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) and the constant for \(\mathrm{N}_{2}\) at \(30^{\circ} \mathrm{C}\) is \(6.0 \times 10^{-4} \mathrm{M} / \mathrm{atm} .\) If the two gases are each present at 1.5 atm pressure, calculate the solubility of each gas.

Short answer

The solubility of helium in water at 30°C and 1.5 atm partial pressure is approximately \(5.55 \times 10^{-4} \mathrm{M}\), and the solubility of nitrogen is approximately \(9.0 \times 10^{-4} \mathrm{M}\).

Step-by-step solution

Identify the given values

We are given the following information: - Henry's law constant for helium (\(k_{He}\)) = \(3.7 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) - Henry's law constant for nitrogen (\(k_{N_2}\)) = \(6.0 \times 10^{-4} \mathrm{M} / \mathrm{atm}\) - Partial pressure for helium and nitrogen (\(P_{He} = P_{N_2}\)) = 1.5 atm

Calculate the solubility of helium

Using the given values for helium, we can plug them into the Henry's law formula to find its solubility: \[C_{He} = k_{He} P_{He}\] \[C_{He} = (3.7 \times 10^{-4} \mathrm{M} / \mathrm{atm}) \times (1.5 \mathrm{atm})\] \[C_{He} = 5.55 \times 10^{-4} \mathrm{M}\] The solubility of helium in water at 30°C and 1.5 atm partial pressure is approximately \(5.55 \times 10^{-4} \mathrm{M}\).

Calculate the solubility of nitrogen

Now, we can use the given values for nitrogen to find its solubility: \[C_{N_2} = k_{N_2} P_{N_2}\] \[C_{N_2} = (6.0 \times 10^{-4} \mathrm{M} / \mathrm{atm}) \times (1.5 \mathrm{atm})\] \[C_{N_2} = 9.0 \times 10^{-4} \mathrm{M}\] The solubility of nitrogen in water at 30°C and 1.5 atm partial pressure is approximately \(9.0 \times 10^{-4} \mathrm{M}\).

Report the results

The solubilities of helium and nitrogen in water at 30°C and 1.5 atm partial pressure are: - Helium: \(5.55 \times 10^{-4} \mathrm{M}\) - Nitrogen: \(9.0 \times 10^{-4} \mathrm{M}\)

Key concepts

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid, forming a solution. It's an essential concept when studying chemical reactions and environments like the ocean or the human bloodstream.
Several factors influence gas solubility, including temperature, pressure, and the nature of the solvent. Typically, as temperature increases, gas solubility decreases because gases expand and escape from the liquid phase.
However, the primary focus here is on how pressure affects solubility, which is beautifully explained by Henry's Law. According to Henry's Law, the amount of gas that dissolves in a liquid is directly proportional to its partial pressure above the liquid. Thus, understanding solubility allows one to predict how gases will behave in various conditions.

Partial Pressure

Partial pressure is an important concept in understanding gas solubility. In a mixture of gases, each gas exerts a pressure independent of the others, called its partial pressure. The total pressure of the gas mixture is the sum of the partial pressures of all the component gases.
Each gas in a solution contributes to the overall pressure based on its proportion in the gas mixture. When working with Henry's Law, knowing the partial pressure of each gas helps determine how much will dissolve in the solvent.
For example, in our exercise with helium and nitrogen gases, both have a partial pressure of 1.5 atm. This information, combined with Henry's Law constant, allows us to calculate their respective solubilities in water.

Henry's Law Constant

Henry's Law is a fundamental principle in chemistry that explains how a gas behaves when in contact with a liquid. The equation is represented as \( C = kP \), where \( C \) is the solubility (concentration) of the gas, \( k \) is Henry's Law constant, and \( P \) is the partial pressure. The constant \( k \) varies depending on the type of gas and temperature.
In our specific exercise, the constants are given for helium \((3.7 \times 10^{-4} \mathrm{M/atm})\) and nitrogen \((6.0 \times 10^{-4} \mathrm{M/atm})\). These constants allow us to determine how much of each gas can dissolve in the solvent under a given partial pressure.
The higher the Henry's Law constant, the more soluble the gas is, everything else being equal. Thus, for a fixed amount of pressure, a gas with a larger \( k \) value will dissolve more in the liquid compared to one with a smaller \( k \), exemplifying the use of Henry's Law in chemical reactions and industrial applications.