Problem 38 - Chemistry The Central Science

Question

The partial pressure of \(\mathrm{O}_{2}\) in air at sea level is \(0.21 \mathrm{~atm} .\) Using the data in Table 13.1 , together with Henry's law, calculate the molar concentration of \(\mathrm{O}_{2}\) in the surface water of a mountain lake saturated with air at \(20^{\circ} \mathrm{C}\) and an atmospheric pressure of 650 torr.

Step-by-Step Solution

Answer

The molar concentration of O₂ in the surface water of the mountain lake at \(20^{\circ}\)C and 650 torr atmospheric pressure can be calculated using Henry's Law and the provided data. First, convert the atmospheric pressure to atm (\(\frac{650}{760} \mathrm{~atm}\)), then find the partial pressure of O₂ at the mountain lake (0.21 × 650/760 atm). Finally, apply Henry's Law using the O₂ constant (\(1.3 \times 10^{-3} \frac{mol}{L \cdot atm}\)) to find the molar concentration of O₂: \((1.3 \times 10^{-3} \frac{mol}{L \cdot atm})\) × (0.21 × 650/760 atm).

Step 1: Convert atmospheric pressure to atm

First, we need to convert the given atmospheric pressure of 650 torr to atm, as Henry's law constant in Table 13.1 is given in atm. We can use the conversion factor 1 atm = 760 torr: Pressure of the mountain lake (atm) = Pressure (torr) / 760 torr

Step 2: Calculate the partial pressure of O₂ at the mountain lake

Next, we need to determine the partial pressure of O₂ at the mountain lake. We can do this by considering that the mole fraction of O₂ in air at sea level is 0.21 atm. We can use the ratio of the pressures and the mole fraction to find the partial pressure of O₂ at the mountain lake: Partial pressure of O₂ at the mountain lake = (Pressure of the mountain lake) × (mole fraction of O₂)

Step 3: Apply Henry's Law

Now, we can use Henry's Law to find the molar concentration of O₂ in the surface water of the mountain lake. Henry's Law states that the concentration of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. So: Molar concentration of O₂ = (Henry's Law constant) × (Partial pressure of O₂ at the mountain lake) From Table 13.1, the Henry's Law constant for O₂ is \(1.3 \times 10^{-3} \frac{mol}{L \cdot atm}\) at \(20^{\circ}\)C.

Step 4: Calculate the molar concentration of O₂

Finally, we can plug in the values we have found into our equation from Step 3: Molar concentration of O₂ = \((1.3 \times 10^{-3} \frac{mol}{L \cdot atm})\) × (Partial pressure of O₂ at the mountain lake) Calculate the result to find the molar concentration of O₂ in the surface water of the mountain lake.

Key Concepts

Partial Pressure

Partial pressure is a fundamental concept in understanding gas mixtures, particularly when dealing with the behavior of gases both above and dissolved in liquids. Imagine you are at a party filled with balloons of different colors, each representing a different gas in a mixture. Just like each balloon occupies a certain space in the room, each gas in a mixture exerts its own pressure, which would be the pressure if it alone occupied the entire volume at the same temperature. That individual pressure is what we call the partial pressure.

The partial pressure of a gas is directly proportional to its mole fraction in the mixture. When we speak of air, which is a mixture of gases, the partial pressure of oxygen (O₂) is a measure of how much oxygen is present and ready to dissolve into liquids such as water. Understanding the partial pressure is crucial, as it forms the backbone of Henry's Law calculations for gas solubility.

Molar Concentration

If you think of a glass of lemonade, the concentration would relate to how much lemon juice is mixed with water. In chemistry, molar concentration (or molarity) is a measure of how many moles of a substance are present in a liter of solution. It is denoted by the unit moles per liter (mol/L).

Molar concentration gives us a clear idea of the strength or intensity of the solution, much like knowing how tangy the lemonade is. Precisely, it tells us the amount of a substance within a specific volume of liquid, which is essential when we want to know how much gas is dissolved in a body of water at equilibrium, per Henry's law.

Solubility of Gases

When you throw a sugar cube into a tea, it dissolves after some time. Similarly, gases like oxygen and carbon dioxide dissolve in liquids. The solubility of gases follows a unique set of rules, heavily influenced by temperature and pressure.

According to Henry's Law, the amount of gas that dissolves in a liquid is directly proportional to its partial pressure above the liquid. The higher the pressure, the more gas will dissolve, until equilibrium is reached. Hence, for divers and aquatic life, the solubility of gases in water is a matter of vital importance as it affects breathing and the underwater ecosystem balance.

Gas-Liquid Equilibrium

Equilibrium is like a perfectly balanced seesaw, with neither side overpowering the other. In the case of gases and liquids, equilibrium refers to a state where the rate of gas molecules entering the liquid is equal to the rate of gas molecules escaping the liquid. This gas-liquid equilibrium is dynamic, meaning the exchange of molecules continues, but the overall amount of gas in the liquid remains constant.

Henry's Law is applicable at this equilibrium point and is crucial in calculating how much gas will remain dissolved under a set of given conditions. This understanding not only helps in industrial processes but also in environmental science, healthcare, and understanding natural occurrences like how fish get oxygen from water.