Problem 94 - Chemistry Matter and Change

Question

Soft Drinks The partial pressure of \(\mathrm{CO}_{2}\) inside a bottle of soft drink is 4.0 atm at \(25^{\circ} \mathrm{C}\) . The solubility of \(\mathrm{CO}_{2}\) is 0.12 \(\mathrm{mol} / \mathrm{L}\) . When the bottle is opened, the partial pressure drops to \(3.0 \times 10^{-4}\) atm. What is the solubility of \(\mathrm{CO}_{2}\) in the open drink? Express your answer in grams per liter.

Step-by-Step Solution

Answer

The solubility of CO2 is approximately 0.000396 g/L.

Step 1: Understand Henry's Law

Henry's Law states that the solubility of a gas in a liquid is directly proportional to its partial pressure above the liquid. Mathematically, it is expressed as: \[C = k_H \times P\]where \(C\) is the solubility of the gas, \(k_H\) is Henry's law constant, and \(P\) is the partial pressure of the gas.

Step 2: Calculate Henry's Law Constant

We know from the initial conditions that when the partial pressure is 4.0 atm, the solubility is 0.12 mol/L. Using Henry's Law, we can solve for the constant \(k_H\):\[k_H = \frac{C}{P} = \frac{0.12}{4.0} = 0.03 \, \frac{\text{mol}}{\text{L}\cdot\text{atm}}\]

Step 3: Find Solubility at Lower Pressure

Now, use the calculated \(k_H\) to find the new solubility \(C\) when the pressure is \(3.0 \times 10^{-4}\) atm:\[C = k_H \times P = 0.03 \times 3.0 \times 10^{-4} = 9.0 \times 10^{-6} \, \text{mol/L}\]

Step 4: Convert Solubility to Grams per Liter

To convert the solubility from mol/L to grams/L, we multiply by the molar mass of \(\mathrm{CO}_2\), which is approximately 44.01 g/mol:\[\text{Solubility in g/L} = 9.0 \times 10^{-6} \, \text{mol/L} \times 44.01 \, \text{g/mol} \approx 3.96 \times 10^{-4} \, \text{g/L}\]

Key Concepts

Solubility

Solubility is an important concept in chemistry that refers to the maximum amount of a substance (solute) that can dissolve in a solvent at a specific temperature and pressure to form a uniform solution. The solubility can vary depending on the nature of the solute and solvent, as well as the conditions of the environment.

• For gases, solubility often depends on pressure. This is depicted in Henry's Law where solubility is directly proportional to the pressure of the gas above the liquid.
• Factors like temperature and the presence of other substances can also impact solubility.

In the exercise, the solubility of carbon dioxide in a soft drink changed due to the alteration of its partial pressure after opening.

Partial Pressure

Partial pressure is a term used to describe the pressure exerted by a single type of gas in a mixture of gases. It's an important factor when dealing with gases dissolved in liquids. In Henry's Law, the concept of partial pressure helps us understand how changes in pressure affect the solubility of a gas.

• Each gas in a mixture contributes to the total pressure in proportion to its abundance.
• In the context of the exercise, the partial pressure of carbon dioxide decreases significantly when a soft drink is opened, which in turn reduces its solubility.

Understanding partial pressure aids in analyzing how gases behave under different conditions, especially in solutions.

Carbon Dioxide Solubility

The solubility of carbon dioxide is particularly important in beverages like soft drinks, where it remains dissolved until the bottle is opened. The dissolution of \( \text{CO}_2 \) gives the drink its fizz.

• The solubility decreases when you open the bottle because the partial pressure of \( \text{CO}_2 \) falls, allowing gas to escape into the atmosphere.
• As seen in the exercise, Henry's Law Illustrates this concept by showing that when the pressure in a closed bottle is 4.0 atm, the drink can hold more \( \text{CO}_2 \) than when it is opened and the pressure drops to \( 3.0 \times 10^{-4} \).
• Solubility is then calculated to show how much carbonation remains in a drink after exposure to air.

This understanding is vital in beverage industries to ensure proper carbonation levels.

Molar Mass of CO2

The molar mass is the sum of the masses of all atoms in a molecule. For carbon dioxide, this is calculated as follows:\[ \text{C} = 12.01 \text{ g/mol}, \text{O} = 16.00 \text{ g/mol} \]Adding these for \( \text{CO}_2 \) gives:\[ 12.01 + (2 \times 16.00) = 44.01 \text{ g/mol} \]

• This value is crucial when converting solubility from mol/L to grams/L, as seen in the exercise where the solubility of carbon dioxide was converted using its molar mass.
• Knowing the molar mass helps in various calculations in chemistry, including stoichiometry and conversion between moles and grams.

In solving problems like this exercise, precise understanding of molar mass enables accurate conversions and results.