Problem 43 - General Chemistry: Principles and Modern Applications

Question

Under an \(\mathrm{O}_{2}(\mathrm{g})\) pressure of \(1.00 \mathrm{atm}, 28.31 \mathrm{mL}\) of \(\mathrm{O}_{2}(\mathrm{g})\) dissolves in \(1.00 \mathrm{L} \mathrm{H}_{2} \mathrm{O}\) at \(25^{\circ} \mathrm{C} .\) What will be the molarity of \(\mathrm{O}_{2}\) in the saturated solution at \(25^{\circ} \mathrm{C}\) when the \(\mathrm{O}_{2}\) pressure is 3.86 atm? (Assume that the solution volume remains at \(1.00 \mathrm{L}\).)

Step-by-Step Solution

Answer

The molarity of O2 in the saturated solution at 25°C and pressure of 3.86 atm would be 3.86 times the initial molarity computed in step 2.

Step 1: Understanding Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to the pressure of that gas above the solution. It is represented by the formula: \(C= k \cdot P\) where C is the concentration of the gas in the solution, P is the pressure of the gas above the solution, and k is the Henry’s law constant.

Step 2: Calculate initial molarity

First, we will calculate the molarity (M) of O2 under 1 atm pressure. Given that 28.31 mL of O2 gas dissolves in 1.00 L of water, remember that molarity is defined as moles of solute per liter of solution. The molar volume of a gas at STP is 22.4 L/mol, so we can calculate moles of O2 as \(moles= \frac{Volume}{Molar\,volume} =\frac{28.31\,mL }{ 22.4 \times 10^3 \,mL/mol}\). Then the molarity M = \( \frac{moles}{ Volume\, of\, solution} \) . Should be computed to know the molarity of O2 in the initial condition.

Step 3: Calculate molarity at higher pressure

According to Henry's Law, if we increase the pressure, the solubility (or molarity) of the gas will increase proportionally. So if the pressure is 3.86 times bigger (3.86 atm), the molarity will also be 3.86 times bigger. So multiply the initial molarity (calculated in step 2) by 3.86 to find the new molarity at 3.86 atm.

Key Concepts

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid. This is a crucial concept in many scientific and industrial processes, such as oxygen dissolving in water for aquatic life. The solubility of a gas depends on several factors including temperature and pressure.

• At higher temperatures, gases generally become less soluble. This is why fizz in a soda becomes less noticeable when the soda warms up.
• Pressure has a direct effect on the solubility of gases, explained by Henry's Law. When the pressure of a gas above a liquid increases, more gas molecules are "pushed" into the liquid, increasing solubility.

Understanding these concepts is essential for fields like environmental science, where gas exchange between bodies of water and the atmosphere is studied.

Molarity Calculation

Molarity is a way to express the concentration of a solute in a solution. It is defined as the number of moles of solute divided by the volume of the solution in liters. The formula for molarity is:

• Molarity (M) = moles of solute / liters of solution

Knowing how to calculate molarity allows you to determine how concentrated a solution is. In our context, we calculate the molarity of the gas (oxygen) that dissolves in water. Given a volume of a gas, we can convert it to moles using its molar volume (22.4 L/mol at STP) and then find the molarity. This process helps in understanding how much oxygen is present in water at different pressures.

Oxygen Dissolution

Oxygen dissolution in water is a natural process that plays a critical role in aquatic ecosystems. The amount of oxygen that dissolves in water influences the health and sustainability of the aquatic environment, as it is vital for the survival of fish and other marine organisms.

• At 1 atmosphere of pressure, a specific volume of oxygen gas will dissolve in water to create a solution. In our example, 28.31 mL of oxygen gas dissolves to form a saturated solution.
• With pressure changes, such as increasing the pressure to 3.86 atm, the amount of oxygen that can dissolve into the water will change. The dissolved oxygen will enhance, directly affecting water habitats.

This process can be influenced by environmental factors, making it an important area of study in environmental science.

Pressure Effects on Solubility

Pressure has a profound effect on the solubility of gases. As described by Henry's Law, the solubility of a gas is directly proportional to the pressure of the gas above the liquid. In simpler terms, if you increase the pressure, more gas will dissolve.

• For example, when the pressure of oxygen is increased from 1 atm to 3.86 atm, more oxygen molecules encounter the surface of the water, resulting in more oxygen dissolving into it.
• This principle is used in carbonated beverages and various industrial applications where control of gas solubility is required.

By understanding pressure effects, scientists and engineers can predict how much of a gas will dissolve in a liquid at different pressures, which is vital for processes such as water treatment and beverage carbonation.