Question

Determine the mole fraction of carbon dioxide \(\left(\mathrm{CO}_{2}\right)\) dissolved in water at the surface of water at \(300 \mathrm{~K}\). The mole fraction of \(\mathrm{CO}_{2}\) in air is \(0.006\), and the local atmosphere pressure is \(100 \mathrm{kPa}\).

Short answer

Answer: The mole fraction of carbon dioxide dissolved in water at the surface of water at 300 K is 3.62 × 10⁻⁵.

Step-by-step solution

Write the formula for Henry's Law

The formula for Henry's Law is given as: \(C = k_H P_A\) where \(C\) is the concentration of the gas in the liquid, \(k_H\) is the Henry's Law constant, \(P_A\) is the partial pressure of the gas in the atmosphere.

Find the partial pressure of CO₂ in air

We are given the local atmospheric pressure \(P_{atm}\) as \(100 \mathrm{kPa}\) and the mole fraction of CO₂ in air \(x_a\) as \(0.006\). Using these values, we can find the partial pressure of CO₂ in air using the following equation: \(P_A = x_a \times P_{atm}\) Substituting the given values, we get: \(P_A = 0.006 \times 100 \mathrm{kPa} = 0.6 \mathrm{kPa}\) The partial pressure of CO₂ in air is \(0.6 \mathrm{kPa}\).

Find the Henry's Law constant for CO₂

The Henry's Law constant for CO₂ dissolved in water at \(300 \mathrm{K}\) is approximately: \(k_H = 3.30 \times 10^{-2} \cfrac{\mathrm{mol}}{\mathrm{L} \cdot \mathrm{atm}}\) Note that we need to convert this constant to the appropriate unit given the partial pressure is in kPa: \(k_H = 3.30 \times 10^{-2} \cfrac{\mathrm{mol}}{\mathrm{L} \cdot \mathrm{atm}} \times \cfrac{101.325 \mathrm{kPa}}{\mathrm{atm}} = 3.35 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L} \cdot \mathrm{kPa}}\)

Calculate the concentration of dissolved CO₂

Now we can use Henry's Law formula and the values we found in Steps 2 and 3 to calculate the concentration of dissolved CO₂: \(C = k_H P_A\) \(C = 3.35 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L} \cdot \mathrm{kPa}} \times 0.6 \mathrm{kPa} = 2.01 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L}}\) The concentration of dissolved CO₂ is \(2.01 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L}}\).

Convert mol/L to mole fraction

Finally, we will convert the concentration of dissolved CO₂ in mol/L to mole fraction. For that, we need to find the total concentration of all solutes in the water. Assuming pure water, the total concentration of water in mol/L \(C_{H_2O}\) can be calculated as: \(C_{H_2O} = \cfrac{\rho_{H_2O}}{M_{H_2O}}\) where \(\rho_{H_2O}\) is the density of water, which is \(1 \cfrac{\mathrm{g}}{\mathrm{mL}} = 1000 \cfrac{\mathrm{g}}{\mathrm{L}}\), and \(M_{H_2O}\) is the molar mass of water, which is \(18.015 \cfrac{\mathrm{g}}{\mathrm{mol}}\). \(C_{H_2O} = \cfrac{1000 \cfrac{\mathrm{g}}{\mathrm{L}}}{18.015 \cfrac{\mathrm{g}}{\mathrm{mol}}} = 55.508 \cfrac{\mathrm{mol}}{\mathrm{L}}\) To find the mole fraction of dissolved CO₂ \(x_{CO_2}\), we can use the equation: \(x_{CO_2} = \cfrac{C_{CO_2}}{C_{CO_2} + C_{H_2O}}\) \(x_{CO_2} = \cfrac{2.01 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L}}}{2.01 \times 10^{-3} \cfrac{\mathrm{mol}}{\mathrm{L}} + 55.508 \cfrac{\mathrm{mol}}{\mathrm{L}}} = 3.62 \times 10^{-5}\) The mole fraction of carbon dioxide dissolved in water at the surface of water at \(300 \mathrm{K}\) is \(3.62 \times 10^{-5}\).