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

Step-by-step solution

Calculate the solubility of helium gas

To calculate the solubility of helium gas in water, we will use the Henry's law formula with the given constant and pressure. The Henry's law constant for helium gas is \(3.7 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}\), and the pressure is 1.5 atm. \[C_{He} = k_{H,He}P_{He}\] \[C_{He} = (3.7 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}) \times (1.5 \, \mathrm{atm})\]

Calculate the solubility of nitrogen gas

To calculate the solubility of nitrogen gas in water, we will use the Henry's law formula with the given constant and pressure. The Henry's law constant for nitrogen gas is \(6.0 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}\), and the pressure is 1.5 atm. \[C_{N_2} = k_{H,N_2}P_{N_2}\] \[C_{N_2} = (6.0 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}) \times (1.5 \, \mathrm{atm})\]

Calculate the final solubilities

Now we will calculate the final solubilities by solving the above equations. For helium gas: \[C_{He} = (3.7 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}) \times (1.5 \, \mathrm{atm}) = 5.55 \times 10^{-4} \, \mathrm{M}\] For nitrogen gas: \[C_{N_2} = (6.0 \times 10^{-4} \, \mathrm{M} / \mathrm{atm}) \times (1.5 \, \mathrm{atm}) = 9.0 \times 10^{-4} \, \mathrm{M}\] So, the solubility of helium gas is \(5.55 \times 10^{-4} \, \mathrm{M}\), and the solubility of nitrogen gas is \(9.0 \times 10^{-4} \, \mathrm{M}\).

Key concepts

Solubility Calculation

When determining the solubility of a gas in a liquid, like helium or nitrogen in water, we rely on a core principle known as Henry's Law. This law allows us to calculate how much gas will dissolve in a liquid based on a constant specific to each gas and the pressure of the gas above the liquid. The essential formula given by Henry's Law is:\[C = k_H \times P\]- **\(C\)** stands for the solubility of the gas in the liquid, expressed in molarity (M).- **\(k_H\)** is the Henry's Law constant, which varies for each gas and temperature; it indicates how soluble the gas is.- **\(P\)** represents the partial pressure of the gas in atmospheres (atm).For helium and nitrogen in this case, we just need to plug the respective constants and their pressures into Henry’s formula to find their solubilities. It is a straightforward multiplication process once these variables are known.

Gas Solubility in Water

The ability of gases to dissolve in water is a fascinating aspect of chemistry that heavily depends on several factors including temperature and the nature of the gas. In this scenario, we have helium and nitrogen, both of which are non-polar gases but behave differently due to their distinct chemical properties and molecular sizes.- **Helium**, being a noble gas, has minimal interaction with water molecules, leading to lower solubility. As calculated using Henry’s Law previously, the solubility of helium in water at 30°C with a partial pressure of 1.5 atm is approximately \(5.55 \times 10^{-4} \, \mathrm{M}\).- **Nitrogen**, on the other hand, is more soluble than helium due to slightly stronger interactions despite being non-polar too. Its solubility under the same conditions is more significant at \(9.0 \times 10^{-4} \, \mathrm{M}\).This difference underscores the behavior variations of gases in water, driven by their specific interactions and molecular sizes. Water, being a polar solvent, generally tends to dissolve gases with a higher polarity better, though both gases are relatively less soluble due to their non-polarity.

Pressure and Concentration Relationship

The relationship between the pressure of a gas and its concentration in a liquid is directly proportional as stipulated by Henry’s Law. This means that as the pressure of the gas over the liquid increases, its solubility or concentration within the liquid also increases.- **Pressure Increase:** If we were to increase the pressure from 1.5 atm to a higher value while keeping the temperature constant, the concentration of the gas in water would increase correspondingly. - **Mathematical Explanation:** Given the formula \(C = k_H \times P\), if the pressure \(P\) is doubled, then the solubility \(C\) also doubles, given that \(k_H\) remains constant for a particular gas at a constant temperature.This concept is crucial when considering scenarios such as carbonated beverages where the pressure of carbon dioxide is increased to dissolve more gas into the liquid. Understanding this principle can help in various applications, from industrial gas-liquid reactions to everyday experiences like opening a fizzy drink.