Question

A carbonated beverage is made by saturating water with carbon dioxide at \(0^{\circ} \mathrm{C}\) and a pressure of 3.0 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 \(\mathrm{CO}_{2}{ }\left(P_{\mathrm{CO}_{2}}=\right.\) \(\left.3.4 \times 10^{-4} \mathrm{~atm}\right)\). 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?

Short answer

The initial concentration of carbon dioxide in the bottle is 0.2307 M, and the final concentration after it has been opened and come to equilibrium with the air is 1.0642 x 10⁻⁵ M.

Step-by-step solution

Calculate initial concentration

To find the concentration of \(\mathrm{CO}_{2}\) in the bottle before it is opened, we will use Henry's law: \[C_1 = k_HP_{\mathrm{CO_{2}}}\] where \(C_1\) is the initial concentration of carbon dioxide, \(k_H\) is the Henry's law constant (0.0769 M/atm at 0℃), and \(P_{\mathrm{CO_{2}}}\) is the pressure of carbon dioxide (3.0 atm). Now, calculate the initial concentration: \(C_1 = 0.0769 \times 3.0 = 0.2307 \mathrm{M}\). Therefore, the initial concentration of carbon dioxide in the bottle is \(0.2307 \mathrm{M}\).

Calculate final concentration

To find the concentration of \(\mathrm{CO}_{2}\) after the bottle is opened, we will use Henry's law again with the atmospheric \(\mathrm{CO}_{2}\) pressure and Henry's law constant at 25℃: \[C_2 = k_H'P_{\mathrm{CO_{2}}'}\] where \(C_2\) is the final concentration of carbon dioxide, \(k_H'\) is the Henry's law constant at 25℃ (0.0313 M/atm), and \(P_{\mathrm{CO_{2}}'}\) is the atmospheric \(\mathrm{CO}_{2}\) pressure (3.4 x 10⁻⁴ atm). Now, calculate the final concentration: \(C_2 = 0.0313 \times 3.4 \times 10^{-4} = 1.0642 \times 10^{-5}\mathrm{M}\). Therefore, the final concentration of carbon dioxide in the bottle is \(1.0642 \times 10^{-5} \mathrm{M}\). (a) The initial concentration of carbon dioxide in the bottle is \(0.2307 \mathrm{M}\). (b) The final concentration of carbon dioxide in the bottle after it has been opened and come to equilibrium with the air is \(1.0642 \times 10^{-5} \mathrm{M}\).