Question

Calculate the solubility of \(\mathrm{O}_{2}\) in water at a partial pressure of \(\mathrm{O}_{2}\) of 120 torr at \(25^{\circ} \mathrm{C}\) . The Henry's law constant for \(\mathrm{O}_{2}\) is \(1.3 \times 10^{-3} \mathrm{mol} / \mathrm{L} \cdot\) atm for Henry's law in the form \(C=k P\) where \(C\) is the gas concentration \((\mathrm{mol} / \mathrm{L})\)

Short answer

The solubility of O₂ in water at a partial pressure of 120 torr and at \(25^{\circ}\)C is approximately \(2.053 \times 10^{-4}\, \text{mol/L}\).

Step-by-step solution

Convert partial pressure from torr to atm

First, we need to convert the given partial pressure of O₂ from torr to atm. Recall the conversion factor: 1 atm = 760 torr. Divide the given partial pressure by 760 to obtain the pressure in atm: \[ P_{O_{2}} = \frac{120 \,\text{torr}}{760\, \text{torr/atm}} \]

Find the partial pressure of O₂ in atm

Calculate the partial pressure in atm: \[ P_{O_{2}} = 0.1579 \,\text{atm}\]

Use Henry's law to calculate the solubility of O₂

Now we will use the Henry's law formula, which is given as: \[ C = kP \] Where \(C\) is the concentration of the gas in mol/L, \(k\) is the Henry's law constant, and \(P\) is the partial pressure of the gas in atm. We are given \(k = 1.3 \times 10^{-3}\, \text{mol/L} \cdot \text{atm}\) and we found \(P = 0.1579\) atm in Step 2. Substitute these values into the Henry's law formula to find the concentration of O₂: \[ C = (1.3 \times 10^{-3}\, \text{mol/L} \cdot \text{atm}) \times 0.1579 \,\text{atm} \]

Calculate the concentration of O₂

Calculate the concentration of O₂ in water: \[ C = 2.053 \times 10^{-4}\, \text{mol/L} \] Thus, the solubility of O₂ in water at a partial pressure of 120 torr and at \(25^{\circ}\)C is approximately \(2.053 \times 10^{-4}\, \text{mol/L}\).

Key concepts

Understanding Partial Pressure

Partial pressure is the pressure that a single gas in a mixture of gases would exert if it took up the entire volume at the same temperature.
When studying gases and their behavior, it's essential to note that each gas in a mixture behaves independently.
Henry's Law, which helps us calculate gas solubility, heavily relies on the concept of partial pressure. In our exercise, we were given the partial pressure of oxygen as 120 torr.
This indicates the pressure that oxygen alone, separate from other gases, would exert if it occupied the same space. To work with this pressure in most gas law equations, we need it in atmospheres (atm) because it's a standard unit. Hence, understanding how to convert torr to atm is vital to applying Henry's law effectively.

Solubility Calculation with Henry's Law

Henry's Law describes how gases dissolve in liquids, stating that the amount of gas dissolved in a liquid is directly proportional to its partial pressure above the liquid.
The relationship is mathematically represented by the formula: \[ C = kP \] Where:

• \(C\) is the concentration of the dissolved gas (mol/L).
• \(k\) is the Henry's law constant, a specific value for each gas.
• \(P\) is the partial pressure of the gas (in atm).

To calculate solubility, one must know the gas's partial pressure and its specific Henry's law constant.
For instance, if the partial pressure of oxygen is 0.1579 atm and the constant \(k\) is \(1.3 \times 10^{-3} \text{ mol/L} \cdot \text{atm}\), the solubility calculation becomes straightforward using the equation.

Evaluating Gas Concentration

Gas concentration refers to how much of a gas is dissolved in a liquid, expressed in molarity (mol/L).
In context, concentration helps us understand how saturated a liquid is with a particular gas. The concentration of gases in liquids is crucial in fields like environmental science and medicine.
For example, understanding oxygen concentration in water bodies helps in assessing water quality for aquatic life. In the exercise we're analyzing, the concentration of oxygen was determined using the Henry’s Law equation.
By substituting the values of \(P\) and \(k\) into the formula \(C = kP\), the oxygen concentration in water was calculated to be approximately \(2.053 \times 10^{-4} \text{ mol/L}\).
This value quantifies how much oxygen is dissolved at that particular pressure.

Explaining Pressure Conversion

Pressure conversion is necessary when dealing with various units of pressure.
Common units include atmospheres (atm), torr, and pascals (Pa).
Converting pressure to a uniform unit, such as atm, streamlines calculations in chemistry. The conversion from torr to atm is especially common, given that 1 atm equals 760 torr.
To convert, divide the pressure value in torr by 760. In practice, like in the given exercise, if oxygen has a partial pressure of 120 torr,

• The conversion to atm is calculated as:
• \( P_{O_{2}} = \frac{120 \text{ torr}}{760 \text{ torr/atm}} = 0.1579 \text{ atm} \).

Performing accurate pressure conversions ensures correct solubility calculations when using the Henry's Law formula.