Question

The solubility of nitrogen in water is \(8.21 \times 10^{-4} \mathrm{~mol} / \mathrm{L}\) at \(0^{\circ} \mathrm{C}\) when the \(\mathrm{N}_{2}\) pressure above water is \(0.790 \mathrm{~atm} .\) Calculate the Henry's law constant for \(\mathrm{N}_{2}\) in units of \(\mathrm{mol} / \mathrm{L} \cdot \mathrm{atm}\) for Henry's law in the form \(C=k P\), where \(C\) is the gas concentration in mol/L. Calculate the solubility of \(\mathrm{N}_{2}\) in water when the partial pressure of nitrogen above water is \(1.10 \mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\).

Short answer

The Henry's law constant for nitrogen in units of mol/L·atm is approximately \(1.04 \times 10^{-3} \mathrm{~mol}/\mathrm{L} \cdot \mathrm{atm}\). The solubility of nitrogen in water when the partial pressure of nitrogen above water is \(1.10 \mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\) is approximately \(1.14 \times 10^{-3} \mathrm{~mol}/\mathrm{L}\).

Step-by-step solution

Write down the given information

We are given: - The solubility of nitrogen in water at \(0^{\circ}\mathrm{C}\) and \(0.790 \mathrm{~atm}\) is \(8.21 \times 10^{-4} \mathrm{~mol}/\mathrm{L}\). - The partial pressure of nitrogen above water is \(1.10 \mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\). We need to determine the Henry's law constant and solubility of nitrogen in water at the new partial pressure.

Use Henry's law formula

According to Henry's law, the solubility of a gas (C) is directly proportional to the partial pressure of the gas (P) when the temperature remains constant. The formula for Henry's law is: \[C = kP\] where \(C\) is the gas concentration in mol/L, \(k\) is the Henry's law constant in units of mol/L·atm, and \(P\) is the partial pressure of the gas in atm.

Calculate the Henry's law constant (k)

Using the given information, we can calculate the Henry's law constant (k) as follows: \[k = \dfrac{C}{P} = \dfrac{8.21 \times 10^{-4} \mathrm{~mol}/\mathrm{L}}{0.790 \mathrm{~atm}}\] Now, compute the value of k: \[k \approx 1.04 \times 10^{-3} \mathrm{~mol}/\mathrm{L} \cdot \mathrm{atm}\]

Calculate the solubility of nitrogen in water at the new partial pressure

According to Henry's law, we have the relationship: \[C = kP\] Now, plug in the values of \(k\) and the new partial pressure (\(P = 1.10 \mathrm{~atm}\)): \[C = (1.04 \times 10^{-3} \mathrm{~mol}/\mathrm{L} \cdot \mathrm{atm})(1.10 \mathrm{~atm})\] Compute the solubility of nitrogen (C) in water at the new partial pressure: \[C \approx 1.14 \times 10^{-3} \mathrm{~mol}/\mathrm{L}\] So, the solubility of nitrogen in water when the partial pressure of nitrogen above water is \(1.10 \mathrm{~atm}\) at \(0^{\circ} \mathrm{C}\) is approximately \(1.14 \times 10^{-3} \mathrm{~mol}/\mathrm{L}\).

Key concepts

Solubility

Solubility refers to the capacity of a substance, such as a gas, to dissolve in a solvent, for instance, water. This capability depends on various factors, including temperature and pressure.
When we talk about the solubility of gases like nitrogen in water, we're examining how much gas can dissolve into the water under specific conditions. At different temperatures or pressure levels, the solubility can vary significantly. Henry's Law helps us understand the behavior of gas solubility relative to changes in pressure. For nitrogen, given conditions of pressure like 0.790 atm, its solubility at 0°C is clearly outlined, providing a structured understanding of how the environment interacts to maintain certain levels of solubility.

Partial Pressure

Partial pressure is the pressure contributed by a single gas in a mixture of gases. Imagine it as the weight or influence each individual gas exerts in the total gaseous environment, encompassing all types of gases present.
For the nitrogen solubility scenario, understanding that the partial pressure is 0.790 atm, and can change to 1.10 atm, guides us in predicting how much nitrogen can dissolve in water. As the partial pressure increases, so does the solubility, assuming the temperature remains constant—this directly follows from Henry's Law. Consequently, knowing the partial pressure assists in calculating the main variable determining gas solubility in the solution, enlightening us on how solubility must adjust with shifts in pressure.

Gas Concentration

Gas concentration denotes how crowded the gas molecules are within a given solution volume. In our context, concentration tells us the quantity of nitrogen dissolved in water, expressed in mol/L.
The given nitrogen concentration was found to be 8.21 x 10^-4 mol/L at an initial partial pressure of 0.790 atm. This straightforward value illustrates how the pressure influences the amount of gas in the water, as concentration directly correlates with solubility, thanks to Henry's Law. Equipped with an initial concentration, we can calculate the new concentration, or solubility, using Henry's Law and any new partial pressure. When partial pressure rises, concentration, or solubility, follows suit, providing a predictable relationship valuable in many scientific and industrial applications.

Henry's Law Constant

Henry's Law constant is a fundamental factor describing a gas's particular preference to dissolve in a specific solvent at a given temperature. We symbolize it as \(k\) in Henry's Law's equation \(C = kP\), where \(C\) represents concentration, and \(P\) is partial pressure.
For nitrogen in water, calculating Henry's Law constant involves taking the quotient of concentration and pressure: \(k = \frac{8.21 \, \times \, 10^{-4} \, \text{mol/L}}{0.790\, \text{atm}} \), resulting in a \(k\) value of approximately 1.04 x 10^-3 mol/L·atm. This constant helps predict how pressure influences solubility: a higher \(k\) signifies a gas more soluble in the solvent under a specific set of conditions. Grasping this concept facilitates understanding gases' behavior in solutions across varying environmental conditions, such as pressure alterations.