Problem 23 - Chemistry: Atoms First

Question

The Henry's law constant for \(\mathrm{CO}_{2}\) is \(3.4 \times 10^{-2} \mathrm{M} / \mathrm{atm}\) at \(25^{\circ} \mathrm{C}\). Assuming ideal solution behavior, what pressure of carbon dioxide is needed to maintain a \(\mathrm{CO}_{2}\) concentration of \(0.10 \mathrm{M}\) in a can of lemon-lime soda?

Step-by-Step Solution

Answer

A pressure of approximately 2.94 atm of carbon dioxide is required.

Step 1: Understand Henry's Law

Henry's Law states that at a constant temperature, the amount of a given gas that dissolves in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid. The formula is expressed as: \[ C = kP \] where \(C\) is the concentration of the gas in the liquid, \(k\) is Henry's law constant for the gas, and \(P\) is the partial pressure of the gas.

Step 2: Rearrange the equation to solve for pressure

To find the pressure, rearrange Henry’s law to solve for \(P\): \[ P = \frac{C}{k} \]

Step 3: Input the known values into the equation

Now, input the concentration of \(\mathrm{CO}_2 (C) = 0.10 \mathrm{M}\) and Henry's law constant for \(\mathrm{CO}_2\) \(k = 3.4 \times 10^{-2} \mathrm{M}/\mathrm{atm}\) into the rearranged formula to calculate the pressure: \[ P = \frac{0.10}{3.4 \times 10^{-2}} = \frac{0.10}{0.034} \]

Step 4: Solve for pressure

Divide the concentration by Henry’s law constant to find the pressure required: \[ P = \frac{0.10}{0.034} \approx 2.94 \mathrm{atm} \] Therefore, a pressure of about 2.94 atm is necessary to maintain a \(\mathrm{CO}_2\) concentration of 0.10 M in the soda.

Key Concepts

Gas Solubility

Gas solubility refers to the ability of a gas to dissolve in a liquid. The solubility of gases is influenced by various factors, such as the nature of the gas and liquid, temperature, and pressure. According to Henry's Law, gas solubility in a liquid at a given temperature is proportional to the partial pressure of the gas above the liquid. The higher the partial pressure, the more gas that will dissolve until equilibrium is reached.

In the context of beverages like soda, gas solubility plays a crucial role. Soda companies use carbon dioxide to carbonate drinks, and they rely on an understanding of solubility to ensure the right amount of fizz. In practice, this means controlling the pressure within the container to adjust the level of carbon dioxide that remains dissolved in the liquid. If the pressure is too low, the drink will go flat, as the gas will escape from the liquid until equilibrium is achieved with the lower pressure.

Partial Pressure

Partial pressure is the pressure that a single gas in a mixture would exert if it occupied the entire volume alone at the same temperature. It's a way of describing how the total pressure of a gas mixture is the sum of the pressures of each individual gas, known as Dalton’s Law of Partial Pressures.

For gases dissolved in liquids, Henry's Law uses the concept of partial pressure to predict and quantify solubility. Each gas's partial pressure in the headspace above the solution is crucial for determining how much gas will dissolve. Going back to the soda can example, the carbon dioxide's partial pressure above the liquid is controlled to ensure that when you open the can, there is enough gas dissolved to give that satisfying fizz. Manufacturers must carefully seal cans to maintain the necessary partial pressure to prevent the carbon dioxide from escaping.

Concentration of Gas in Liquid

The concentration of gas in liquid is measured in molarity (M), which denotes moles of gas per liter of liquid. Henry's Law provides a direct mathematical relationship to predict the concentration of a dissolved gas, asserting that it is directly proportional to the partial pressure of that gas above the liquid. This allows us to determine the amount of a gas that will dissolve in a liquid under specific pressure conditions.

In practical terms, if we refer to the original exercise involving a soda can, the desired concentration of CO2 is specified at 0.10 M. To maintain this concentration, we use Henry’s Law to calculate the required partial pressure of CO2. Remember that the constant in Henry's Law is specific to each gas and depends on temperature. Adjusting the pressure over the liquid can change the concentration of the dissolved gas—a principle that has wide applications in various technologies, including carbonated beverages, scuba diving, and even in the medical field for administering gases to patients.