Problem 29 - Chemistry: Principles and Reactions

Question

A carbonated beverage is made by saturating water with carbon dioxide at \(0^{\circ} \mathrm{C}\) and a pressure of \(3.0 \mathrm{~atm}\). The bottle is then opened at room temperature \(\left(25^{\circ} \mathrm{C}\right)\), and comes to equilibrium with air in the room containing \(\mathrm{CO}_{2}\left(P_{\mathrm{CO}_{2}}=3.4 \times 10^{-4} \mathrm{~atm}\right)\). The Henry's law constant for the solubility of \(\mathrm{CO}_{2}\) in water is \(0.0769\) M/atm at \(0^{\circ} \mathrm{C}\) and \(0.0313 \mathrm{M} / \mathrm{atm}\) at \(25^{\circ} \mathrm{C}\). (a) What is the concentration of carbon dioxide in the bottle before it is opened? (b) What is the concentration of carbon dioxide in the bottle after it has been opened and come to equilibrium with the air?

Step-by-Step Solution

Answer

(a) The concentration of carbon dioxide in the bottle before it is opened (at \(0^{\circ} \mathrm{C}\)) is approximately 0.231 M. (b) The concentration of carbon dioxide in the bottle after it has been opened and come to equilibrium with the air (at \(25^{\circ} \mathrm{C}\)) is approximately \(1.06 \times 10^{-5} \mathrm{~M}\).

Step 1: Calculate \(\mathrm{CO}_{2}\) concentration at \(0^{\circ} \mathrm{C}\) before opening

Using Henry's Law, we can calculate the concentration of \(\mathrm{CO}_{2}\) at initial conditions (0°C under 3 atm pressure). Henry's Law states that the concentration of a gas in a liquid is proportional to the pressure of the gas above the liquid. The relationship can be written as: $$C = k \times P$$ Where, \(C\) is the concentration of the gas in the liquid, in this case, \(\mathrm{CO}_{2}\), \(k\) is the Henry's law constant for the solubility of the gas in the liquid, and \(P\) is the partial pressure of the gas. Henry's law constant for the solubility of \(\mathrm{CO}_{2}\) in water at \(0^{\circ} \mathrm{C}\) is \(0.0769\) M/atm. The \(\mathrm{CO}_{2}\) pressure in the bottle is 3 atm. Now, let's find concentration \((C_{0})\) of the carbon dioxide before opening the bottle: $$C_{0}=k_{0} \times P_{0}$$ $$C_{0}=0.0769 \mathrm{~M/atm} \times 3.0 \mathrm{~atm}$$

Step 2: Calculate \(\mathrm{CO}_{2}\) concentration at \(25^{\circ} \mathrm{C}\) after opening

After the bottle has been opened and equilibrated with the room air containing CO2, we can use Henry's Law again to calculate the concentration of \(\mathrm{CO}_{2}\) at these conditions. The temperature is now \(25^{\circ} \mathrm{C}\), so we use the Henry's law constant for the solubility of \(\mathrm{CO}_{2}\) in water at this temperature: \(0.0313\) M/atm. The partial pressure of \(\mathrm{CO}_{2}\) in room air is given as \(3.4 \times 10^{-4}\) atm. Now, let's find concentration \((C_{25})\) of the carbon dioxide after opening the bottle: $$C_{25}=k_{25} \times P_{\mathrm{CO}_{2}}$$ $$C_{25}=0.0313 \mathrm{~M/atm} \times 3.4 \times 10^{-4} \mathrm{~atm}$$ Now, we can summarize our results.

Step 3: Answer (a)

The concentration of carbon dioxide in the bottle before it is opened (at \(0^{\circ} \mathrm{C}\)) is: $$C_{0}=0.0769 \mathrm{~M/atm} \times 3.0 \mathrm{~atm} \approx 0.231 \mathrm{~M}$$

Step 4: Answer (b)

The concentration of carbon dioxide in the bottle after it has been opened and come to equilibrium with the air (at \(25^{\circ} \mathrm{C}\)) is: $$C_{25}=0.0313 \mathrm{~M/atm} \times 3.4 \times 10^{-4} \mathrm{~atm} \approx 1.06 \times 10^{-5} \mathrm{~M}$$

Key Concepts

Solubility of Gases

Gas solubility refers to the capacity of a given volume of liquid to hold a particular gas at a specified temperature and pressure. This solubility can be drastically affected by temperature — generally, as the temperature increases, the solubility of a gas in a liquid decreases. This is why a cold carbonated drink releases fewer bubbles than a warm one.

Henry's Law provides a useful way to quantify this relationship, giving a constant that varies with temperature changes. The constant essentially relates the pressure of the gas above the liquid to its concentration within the liquid, establishing a linear relationship. This linear relationship is why when you increase the pressure of carbon dioxide in the carbonation process, more gas dissolves until the system reaches equilibrium.

Gas-Liquid Equilibrium

When we talk about gas-liquid equilibrium, we're discussing the point at which the rate of gas molecules entering the liquid phase is equal to the rate of gas molecules leaving the liquid phase. This equilibrium state means no net change in the concentration of the gas in the liquid over time, assuming constant temperature and pressure.

In the case of sparkling beverages like the example given, carbon dioxide is infused into the liquid under high pressure. The high pressure maintains a higher concentration of the gas dissolved in the liquid than would be found in equilibrium with the gas at atmospheric pressure. Once the bottle is opened, the gas's partial pressure drops dramatically, and the drink begins to degas as it approaches a new equilibrium state with the surrounding environment.

Concentration Calculations

Understanding how to calculate concentrations is critical in chemistry, as it relates to the amount of substance contained in a particular volume. Calculations using Henry's Law are straightforward: multiply the Henry's Law constant by the partial pressure of the gas.

However, this assumes the constant itself doesn't change. Since the constant can vary with temperature, you must ensure you're using the appropriate value for your calculations, as shown in the text's step-by-step solution for different temperatures. Correct concentration calculations allow us to predict and control the properties of solutions, vital for many scientific and industrial processes.