Problem 114 - Chemistry: The Central Science in SI Units, Global Edition

Question

What is the \(\mathrm{pH}\) at \(25^{\circ} \mathrm{C}\) of water saturated with \(\mathrm{CO}_{2}\) at a partial pressure of \(111.5 \mathrm{kPa}\) ? The Henry's law constant for \(\mathrm{CO}_{2}\) at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-4} \mathrm{~mol} / \mathrm{L}-\mathrm{kPa}\).

Step-by-Step Solution

Answer

The pH of water saturated with CO₂ at a partial pressure of 111.5 kPa and a temperature of 25°C is approximately 4.44.

Step 1: Determine the dissolved CO₂ concentration using Henry's Law constant

Using Henry's Law constant, we can calculate the concentration of CO₂ dissolved in water: \[ C_{CO_{2}} = k_{H} \times P \] where \(C_{CO_{2}}\) is the concentration of CO₂, \(k_{H}\) is Henry's Law constant, and \(P\) is the partial pressure of CO₂. \[ C_{CO_{2}} = (3.1 \times 10^{-4} \, \mathrm{mol/L/kPa}) \times (111.5 \, \mathrm{kPa}) \] \[ C_{CO_{2}} = 0.034565 \, \mathrm{mol/L} \]

Step 2: Write the dissolution and formation equation of carbonic acid

The dissolution of CO₂ in water and the subsequent formation of carbonic acid (H₂CO₃) can be represented as follows: \[ CO_{2} \, (g) + H_{2}O \, (l) \rightleftharpoons H_{2}CO_{3} \, (aq) \]

Step 3: Write the ionization equation of carbonic acid

The ionization of carbonic acid (H₂CO₃) into hydrogen ions (H⁺) and bicarbonate ions (HCO₃⁻) can be represented as follows: \[ H_{2}CO_{3} \, (aq) \rightleftharpoons H^{+} \, (aq) + HCO_{3}^{-} \, (aq) \]

Step 4: Calculate the hydrogen ion concentration using the equilibrium constant

The equilibrium constant (Ka1) for the ionization of carbonic acid is \(4.45 \times 10^{-7}\). Using this equilibrium constant, we can calculate the concentration of hydrogen ions (H⁺): \[ K_{a1} = \frac{[H^{+}][HCO_{3}^{-}]}{[H_{2}CO_{3}]} \] Assuming that the concentration of H⁺ and HCO₃⁻ ions is the same (since they both come from the ionization of one molecule of H₂CO₃) and using the concentration of H₂CO₃ obtained in step 1, we can solve for the H⁺ concentration: \[ 4.45 \times 10^{-7} = \frac{[H^{+}]^{2}}{0.034565} \] \[ [H^{+}]^{2} = (4.45 \times 10^{-7}) \times 0.034565 \] \[ [H^{+}] = \sqrt{(4.45 \times 10^{-7}) \times 0.034565} = 3.59 \times 10^{-5} \, \mathrm{mol/L} \]

Step 5: Calculate the pH

Now that we have the concentration of hydrogen ions (H⁺), we can calculate the pH using the following equation: \[ pH = -\log{[H^{+}]} \] \[ pH = -\log{(3.59 \times 10^{-5})} = 4.44 \] So, the pH of water saturated with CO₂ at a partial pressure of 111.5 kPa and a temperature of 25°C is approximately 4.44.

Key Concepts

Henry's Law

When a gas is in contact with a liquid, it can dissolve into the liquid. The amount of gas that dissolves is directly proportional to its partial pressure above the liquid. This concept is defined by Henry's Law. For a particular gas, we express this with the formula:

• \[ C_{gas} = k_H \times P \]

Here, \(C_{gas}\) is the concentration of gas in the liquid, \(k_H\) is Henry's Law constant, and \(P\) is the partial pressure of the gas.
This principle tells us that a higher pressure results in more gas being dissolved. In the context of our exercise, this means knowing the partial pressure of carbon dioxide \(CO_2\) allows us to calculate how much of it is dissolved in water.

Carbonic Acid

Carbonic acid (\(H_2CO_3\)) is a weak acid that forms when carbon dioxide (\(CO_2\)) dissolves in water. The process can be described by the chemical reaction:

• \[ CO_2 (g) + H_2O (l) \rightleftharpoons H_2CO_3 (aq) \]

This equilibrium shows that carbonic acid exists naturally as a result of \(CO_2\) in the atmosphere interacting with rainwater. Carbonic acid plays a key role in maintaining the pH balance in aqueous solutions and in natural systems like our blood and the oceans.
Understanding this reaction is essential as it sets the stage for further reactions when assessing the pH levels using carbonate compounds.

pH Calculation

The pH of a solution is a measure of how acidic or basic it is. Specifically, it is the negative logarithm of the hydrogen ion concentration:

• \[ pH = -\log{[H^+]} \]

In our specific exercise, we used Henry's Law to find the concentration of dissolved \(CO_2\), which then allowed us to calculate the concentration of hydrogen ions using the ionization of carbonic acid. Calculating the pH involves substituting this hydrogen ion concentration into the pH formula above.
This calculation provides insight into the acidity of rainwater or the ocean, both influenced by \(CO_2\) absorption.

Equilibrium Constant

The equilibrium constant (\(K_a\)) expresses the extent of dissociation or ionization of a solute in a solution. For carbonic acid, the equilibrium constant for its first ionization is significant:

• \[ K_{a1} = \frac{[H^+][HCO_3^-]}{[H_2CO_3]} \]

A higher \(K_a\) value indicates a stronger acid that dissociates more in solution. In our exercise, the given \(K_{a1}\) for \(H_2CO_3\) allowed us to find the \([H^+]\) concentration from the initial concentration of carbonic acid.
This process illustrates the relationship between reactants and products in a reversible chemical reaction, providing a quantitative measure of their equilibrium.