Question

\(1 \mathrm{~kg}\) of water under a nitrogen pressure of 1 atmosphere dissolves \(0.02 \mathrm{gm}\) of nitrogen at 293K. Calculate Henry's law constant : (a) \(7.7 \times 10^{4}\) atm (b) \(7.7 \times 10^{3}\) atm (c) \(2 \times 10^{-5} \mathrm{~atm}\) (d) \(2 \times 10^{-2} \mathrm{~atm}\)

Short answer

Henry's law constant $$ k_H $$is approximately $$ 7.7 \times 10^{4}~atm $$.

Step-by-step solution

Understand Henry's Law

According to Henry's law, the amount of gas that dissolves in a liquid at a given temperature is directly proportional to the partial pressure of the gas above the liquid. The mathematical representation of Henry's law is given by: $$ P = k_H \times C $$where,* P is the partial pressure of the gas,* \(k_H\) is Henry's law constant, and* C is the concentration of the gas in the liquid.

Conversion of Mass into Moles for Nitrogen

First convert the amount of dissolved nitrogen from grams to moles to get the concentration (C) since Henry's law involves concentration in moles per volume (and we can assume 1 kg of water to have a volume of 1 L for dilute solutions). The molar mass of nitrogen (N2) is 28.0134 g/mol. The number of moles of nitrogen is calculated using the formula: $$ n = \frac{mass}{molar~mass} = \frac{0.02~g}{28.0134~g/mol} $$

Calculating the Concentration of Nitrogen

Use the number of moles from the previous step to find the concentration of nitrogen in water. Since the volume of water is 1 kg (or 1 L assuming the density of water to be 1 g/mL), the concentration C is equal to the number of moles per liter. Thus, C becomes equal to the moles of nitrogen calculated in the previous step.

Applying Henry's Law to Find Henry's Constant

The partial pressure of nitrogen (P) given is 1 atm. Now use Henry's law equation to solve for Henry's law constant ($$ k_H = \frac{P}{C} $$Plug in the values of P and C obtained from the previous steps to calculate $$ k_H $$

Key concepts

Henry's Law

When it comes to understanding how gases dissolve in liquids, Henry's Law is a fundamental principle in chemistry. It states that at a constant temperature, the amount of a given gas that dissolves in a liquid is directly proportional to the partial pressure of that gas above the liquid. This means that if you increase the gas pressure, more gas will dissolve until a dynamic equilibrium is reached.

The formula for Henry's Law is: \[ P = k_H \times C \] where \(P\) represents the partial pressure of the gas, \(k_H\) is Henry's law constant unique for each gas-liquid pair at a specific temperature, and \(C\) is the concentration of the dissolved gas in the liquid phase. In essence, this law helps us calculate how much of a gas will be dissolved in a liquid under various pressures, which is crucial for applications ranging from carbonated beverages to the medical field, where the oxygenation of blood is vital.

Partial Pressure

The concept of partial pressure is essential when dealing with mixtures of gases. It is defined as the pressure that a single gas component in a mixture of gases would exert if it occupied the entire volume alone at the same temperature. To understand this better, imagine you have a container with both oxygen and nitrogen gases. The total pressure inside the container is the sum of the pressures each gas would exert if it were the only gas present.

To relate this to Henry's Law, we focus on the pressure of the specific gas that is dissolving into the liquid. When calculating Henry's law constant, we use the partial pressure of the gas of interest because it drives the dissolving process. That is why in our exercise, we only consider the pressure of nitrogen when calculating its solubility in water.

Gas Solubility

Gas solubility indicates how much of a particular gas can be dissolved in a liquid at a given temperature and pressure. Solubility can vary widely depending on the nature of the gas and the liquid. For example, carbon dioxide is more soluble in water than nitrogen.

The solubility of gases also decreases with increasing temperature because higher kinetic energy causes gas molecules to escape more easily from the liquid phase. The amount of gas that dissolves is also influenced by the presence of other solutes and the pH of the liquid. An understanding of gas solubility is crucial in a variety of fields, from environmental science to the beverage industry.

Molar Mass

Every substance has a characteristic molar mass, which is the mass of one mole of its molecules or atoms, typically expressed in grams per mole (g/mol). For elements, the molar mass in grams per mole is numerically equal to the atomic weight of the element as listed on the periodic table. The molar mass is an essential factor in stoichiometry, as it allows for the conversion between mass and number of moles of a substance.

In the context of Henry's law, molar mass enables us to convert the mass of a gas into moles, which is necessary for calculating the gas's concentration in the liquid. This step is critical, as Henry's law requires the concentration of the gas to be in moles per liter (mol/L).

Concentration Calculations

Concentration describes the amount of a substance in a given volume of solution and can be expressed in various ways, including moles per liter (Molarity), grams per liter, and percentages. For gases dissolved in liquids, concentration is typically given in terms of molarity because it provides a direct measure of how many moles of gas are present in one liter of solution.

Understanding how to perform concentration calculations is essential for using Henry's law. Once you know the mass of a gas dissolved in a liquid, you convert it to moles using the molar mass. Then you can find the molarity by dividing the number of moles of the gas by the volume of the solution in liters. This is the concentration value \(C\) used in the Henry's law equation to calculate the gas's solubility under a set partial pressure.