Question

The Henry's law constant for \(\mathrm{CO}_{2}\) in water at \(25^{\circ} \mathrm{C}\) $$ \text { is } 3.1 \times 10^{-2} M \mathrm{~atm}^{-1} $$ (a) What is the solubility of \(\mathrm{CO}_{2}\) in water at this temperature if the solution is in contact with air at normal atmospheric pressure? (b) Assume that all of this \(\mathrm{CO}_{2}\) is in the form of \(\mathrm{H}_{2} \mathrm{CO}_{3}\) produced by the reaction between \(\mathrm{CO}_{2}\) and \(\mathrm{H}_{2} \mathrm{O}:\) $$ \mathrm{CO}_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{CO}_{3}(a q) $$ What is the \(\mathrm{pH}\) of this solution?

Short answer

The solubility of CO2 in water at 25°C and normal atmospheric pressure is \(3.1 \times 10^{-2} M\). Assuming all dissolved CO2 is in the form of H2CO3, the pH of the solution can be calculated using the formula: \(pH = -\log_{10}(\sqrt{4.5 \times 10^{-7} \cdot (3.1 \times 10^{-2})})\) After calculation, the pH of the resulting solution is approximately 4.3.

Step-by-step solution

Use Henry's Law to calculate the solubility of CO2

Henry's Law states that the concentration of a gas in a liquid is directly proportional to the partial pressure of the gas. Mathematically, it can be written as: \(C = k_H P\) Where - \(C\) is the concentration of the gas in the liquid (in this case CO2 in water) - \(k_H\) is the Henry's Law constant (\(3.1 \times 10^{-2} M atm^{-1}\)) - \(P\) is the partial pressure of the gas (we'll have to use normal atmospheric pressure, 1 atm) Now, plug in the given values to find the concentration (solubility) of CO2: \(C = (3.1 \times 10^{-2} M\cdot atm^{-1}) \cdot (1 atm)\)

Calculate the solubility of CO2

Multiply the values from step 1 to find the solubility of CO2: \(C = 3.1 \times 10^{-2} M\) This means, under the given temperature and pressure, the solubility of CO2 in water is \(3.1 \times 10^{-2} M\).

Assume all CO2 is in the form of H2CO3

As the CO2 dissolves, it reacts with water to form carbonic acid, H2CO3, according to the reaction: \(\mathrm{CO_2}(aq) + \mathrm{H_2O}(l) \longrightarrow \mathrm{H_2CO_3}(aq)\) Since we assume all the dissolved CO2 is in the form of H2CO3, the concentration of H2CO3 is equal to the solubility of CO2: [CO2] = [H2CO3] = \(3.1 \times 10^{-2} M\)

Calculate the pH using the H2CO3 concentration

Carbonic acid, H2CO3, is a weak acid and will dissociate partially in water: \(\mathrm{H_{2}CO_{3}}(a q) \rightleftharpoons \mathrm{H^{+}}(a q)+\mathrm{HCO_{3}^{-}}(a q)\) Since H2CO3 is a weak acid, its dissociation is characterized by its acid dissociation constant (Ka), which is approximately \(4.5 \times 10^{-7}\) for carbonic acid. Now, we apply the formula for acid dissociation: \(K_{a}=\frac{[\mathrm{H^+}][\mathrm{HCO_3^-}]}{[\mathrm{H_2CO_3}]}\) As we assume that initially, only H2CO3 is present in the solution, we can write the following relationship: \(K_{a}=\frac{x \cdot x}{(3.1 \times 10^{-2}-x)}\) Where x is the concentration of H+ ions. Since Ka is very small, we can assume that x is much smaller than\(3.1 \times 10^{-2}\); then we can simplify the equation to: \(K_{a}=x^2 / 3.1 \times 10^{-2}\) Because Ka = \(4.5 \times 10^{-7}\), we get: \(x^2 = 4.5 \times 10^{-7} \cdot (3.1 \times 10^{-2})\) Now, we can find the concentration of H+ ions by solving for x.

Find the concentration of H+ ions and calculate pH

Solving for x, we have: \(x = \sqrt{4.5 \times 10^{-7} \cdot (3.1 \times 10^{-2})}\) Finally, to find the pH of the solution, we apply the pH formula: pH = \( -\log_{10}[H^{+}]\) pH = \( -\log_{10}(\sqrt{4.5 \times 10^{-7} \cdot (3.1 \times 10^{-2})})\) Calculate the pH value to get the final answer.

Key concepts

CO2 solubility

Understanding the solubility of carbon dioxide (CO2) in water is essential for various scientific and environmental processes. The solubility refers to how much CO2 can be dissolved in water at a specific temperature and pressure. Henry's Law provides a straightforward way to determine this. The law states that the concentration of a gas dissolved in a liquid is directly proportional to the gas's partial pressure. This relationship is described in the formula: \(C = k_H P\).
Thus, the Henry's Law constant \(k_H\) is crucial here. For CO2 at 25°C, it is given as \(3.1 \times 10^{-2} M \cdot atm^{-1}\). Similarly, normal atmospheric pressure is approximately 1 atm.
By substituting these values into Henry's equation, we can compute the solubility of CO2 in water: \(C = (3.1 \times 10^{-2} M \cdot atm^{-1}) \cdot (1 atm)\). The solubility, in this case, is \(3.1 \times 10^{-2} M\).
This calculation shows how much CO2 can be dissolved under these specific conditions, providing insights into aquatic carbon levels and related phenomena such as carbon capture processes.

Carbonic acid formation

When CO2 dissolves in water, it reacts to form carbonic acid, \(\mathrm{H}_2\mathrm{CO}_3\). This is an important reaction because it influences the acidity of natural waters.
The chemical reaction is as follows: \(\mathrm{CO_2}(aq) + \mathrm{H_2O}(l) \rightarrow \mathrm{H_2CO_3}(aq)\).
The assumption in this exercise is that all dissolved CO2 is converted into carbonic acid, meaning that the concentration of \(\mathrm{H}_2\mathrm{CO}_3\) is equal to the solubility of CO2, which is \(3.1 \times 10^{-2} M\).
This conversion is important because carbonic acid is a weak acid, which partially ionizes in water, affecting the water's pH. The understanding of this process is essential for studying the carbon cycle and its effects on ecosystems, as it highlights how CO2 can alter water chemistry.

pH calculation

Calculating the pH of a solution involves understanding the concentration of hydrogen ions, \([H^+]\), in the solution. For carbonic acid, the dissociation into hydrogen ions can be defined by its acid dissociation constant, \(K_a\).
In this case, \(\mathrm{H}_2\mathrm{CO}_3\) dissociates as follows: \(\mathrm{H}_{2}\mathrm{CO}_{3}(aq) \rightleftharpoons \mathrm{H^{+}}(aq) + \mathrm{HCO_{3}^{-}}(aq)\). The acid dissociation constant \(K_a\) for carbonic acid is approximately \(4.5 \times 10^{-7}\).
We use the expression: \(K_a = \frac{[\mathrm{H^+}][\mathrm{HCO_3^-}]}{[\mathrm{H_2CO_3}]}\). Assuming \(x\), the concentration of \([H^+]\), is much smaller than \(3.1 \times 10^{-2} M\), the equation simplifies to \(K_a = \frac{x^2}{3.1 \times 10^{-2}}\).
Solving for \(x\) gives us \(x = \sqrt{4.5 \times 10^{-7} \cdot (3.1 \times 10^{-2})}\). Finally, the pH is calculated using the formula \(\text{pH} = -\log_{10}[H^{+}]\), providing the solution's acidity level. This calculation is fundamental in understanding the impact of CO2 on water acidity and broader environmental implications.