Question

The solubility of nitrous oxide is \(0.12 \mathrm{~g} / 100 \mathrm{~g}\) water at \(20^{\circ} \mathrm{C}\) and 1.00 atm. What is the partial pressure required to dissolve \(0.55 \mathrm{~g}\) of the gas in \(100 \mathrm{~g}\) of water at \(20^{\circ} \mathrm{C} ?\)

Short answer

The partial pressure required is approximately 4.58 atm.

Step-by-step solution

Understand the Given Data

We are given the solubility of nitrous oxide as \(0.12 \text{ g / 100 g water}\) at \(20^{\circ}\text{C}\) and \(1.00 \text{ atm}\). We need to find the partial pressure that would allow \(0.55 \text{ g}\) to dissolve in \(100 \text{ g}\) of water.

Apply Henry's Law

Henry's Law states that \(\text{solubility} = k_H \times \text{pressure}\). Where \(k_H\) is the Henry's Law constant, which we can determine from the initial solubility conditions. \(k_H = \frac{0.12}{1.00} = 0.12 \text{ g/atm} \).

Set Up the Equation for New Solubility

Using the same formula \(\text{solubility} = k_H \times \text{pressure}\), we need to find the pressure for the new solubility condition: \(0.55 \text{ g/100 g water} = 0.12 \times p\).

Solve for New Pressure

Rearrange the equation to solve for \(p\): \(p = \frac{0.55}{0.12} = 4.5833 \text{ atm}\). Round this to two decimal places to get \(p \approx 4.58 \text{ atm}\).

Key concepts

Solubility

Solubility is crucial to understanding how substances mix and dissolve in a solution. It measures how much of a substance can dissolve in a given amount of solvent under specific conditions. In the context of our exercise, solubility tells us the amount of nitrous oxide that can dissolve in water at a given temperature and pressure. For example, at 20°C and 1.00 atm, nitrous oxide's solubility is 0.12 grams per 100 grams of water. This means under these conditions, 0.12 grams of nitrous oxide will dissolve completely in 100 grams of water. When conditions change, like temperature or pressure, solubility can change too. Knowing solubility helps predict how a gas or solid will behave in different solutions.

Partial Pressure

Partial pressure is a concept used in the context of gas mixtures. It refers to the pressure that a single gas in a mixture would exert if it occupied the entire volume alone. This is important because it's directly related to how a gas dissolves in a liquid. In the problem, partial pressure allows us to determine the pressure needed to dissolve a specific amount of gas, like nitrous oxide, in water. Using Henry's Law, we understand that the gas solubility in a liquid is proportional to its partial pressure above the liquid. Hence, knowing the required solubility of 0.55 grams, we can calculate the partial pressure needed to achieve this solubility level.

Solubility Constant

The solubility constant, often represented as a part of Henry's Law, is crucial for quantifying the relationship between solubility and pressure. People's understanding can be deepened by this concept as it shows how much of the gas will dissolve in the liquid at a given pressure.In the exercise, we determine the solubility constant, denoted as \( k_H \), using the initial solubility conditions: \( k_H = \frac{0.12}{1.00} = 0.12 \, \text{g/atm} \). The constant helps us to set up equations to predict new solubility conditions when pressure changes. Essentially, the solubility constant provides a consistent baseline to relate solubility with varying pressures through different scenarios.

Gas Solubility

Gas solubility is a key part of Henry's Law, referring to how a gas dissolves in a liquid. This principle explains how gases, like nitrous oxide, spread into a liquid until equilibrium is reached. In the exercise, we're interested in dissolving more nitrous oxide in water than usual. Therefore, we need to calculate a new pressure that will allow 0.55 grams of nitrous oxide to dissolve in 100 grams of water. To achieve this, we rearrange and solve the Henry’s Law equation for pressure, knowing the solubility and solubility constant. The result, requiring a partial pressure of about 4.58 atm, reflects the higher pressure needed to dissolve more gas than under the original conditions. Understanding gas solubility involves conceptualizing how gases interact with liquids and respond to temperature and pressure changes.