Problem 89 - Chemistry The Central Science

Question

The Henry's law constant for \(\mathrm{CO}_{2}\) in water at \(25^{\circ} \mathrm{C}\) is \(3.1 \times 10^{-2} \mathrm{Matm}^{-1}\) . (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 pH of this solution?

Step-by-Step Solution

Answer

The solubility of CO₂ in water at 25°C and normal atmospheric pressure is 0.031 M. Assuming that all of the CO₂ is in the form of H₂CO₃ formed by the reaction between CO₂ and H₂O, the pH of the solution is approximately 1.51.

Step 1: Recall Henry's Law

Henry's Law states that the solubility of a gas in a liquid is proportional to its partial pressure above the liquid. The equation for Henry's Law is given by: C = KH * P Where C is the concentration of the gas in solution (in M, mol/L), KH is the Henry's Law constant (in M*atm), and P is the partial pressure of the gas above the solution (in atm). In this case, we are given the KH for CO₂ in water at 25°C (3.1 x 10⁻² M*atm⁻¹) and the atmospheric pressure (1 atm). We will use these values to find the solubility (concentration) of CO₂ in water.

Step 2: Calculate the concentration of CO₂ in water

Using the formula for Henry's Law, we can calculate the concentration of CO₂ in water as follows: C = KH * P C = (3.1 x 10⁻² M*atm⁻¹) * (1 atm) C = 0.031 M So, the solubility of CO₂ in water at 25°C and normal atmospheric pressure is 0.031 M. #b) pH of the solution#

Step 1: Reaction of CO₂ with H₂O to form H₂CO₃

We are given that all of the CO₂ in the solution is in the form of H₂CO₃ produced by the reaction between CO₂ and H₂O: CO₂(aq) + H₂O(l) → H₂CO₃(aq) Since we found the concentration of CO₂ in water (0.031 M), we can assume that the concentration of H₂CO₃ in the solution is also 0.031 M.

Step 2: Determine pH from H₂CO₃ concentration

Although H₂CO₃ is a weak acid, we can simplify the problem by assuming that it fully dissociates into H⁺ and HCO₃⁻ ions. Therefore, the concentration of H⁺ ions in the solution is equal to the concentration of H₂CO₃ (0.031 M). To find the pH of the solution, use the equation: pH = -log₁₀[H⁺]

Step 3: Calculate the pH of the solution

Plug in the concentration of H⁺ ions into the pH equation: pH = -log₁₀(0.031) pH ≈ 1.51 The pH of the solution is approximately 1.51.

Key Concepts

Understanding Henry's Law and Gas Solubility

Henry's Law is a fundamental principle in chemistry that helps to predict the solubility of gases in liquids. According to this law, at a constant temperature, the solubility of a gas in a liquid is directly proportional to the partial pressure of the gas above the liquid. The mathematical expression of Henry's Law is given by the equation:
\(C = K_H \cdot P\)
where \(C\) represents the solubility or concentration of the gas in the liquid (in moles per liter, M), \(K_H\) is Henry's Law constant for that particular gas in that particular liquid (in M\(\cdot\)atm\(^{-1}\)), and \(P\) is the partial pressure of the gas in atmospheres (atm).
For example, if we have carbon dioxide (\(\mathrm{CO}_2\)) dissolved in water at a known temperature with a given partial pressure, we can use Henry's Law to calculate its concentration in water. This fundamental understanding is crucial when addressing questions such as the impact of increased carbon dioxide levels on oceanic solubility and the resulting effects on marine life.

Chemical Equilibrium in Reactions Involving Gases

Chemical equilibrium is a state in a chemical reaction where the rate of the forward reaction equals the rate of the reverse reaction, resulting in no net change in the concentrations of reactants and products over time. It plays a crucial role when dealing with reactions involving gases, such as the formation of carbonic acid (\(\mathrm{H}_2\mathrm{CO}_3\)) from the dissolution of carbon dioxide (\(\mathrm{CO}_2\)) in water (\(\mathrm{H}_2\mathrm{O}\)).
This reaction is important to consider when discussing the solubility of gases:
\[\mathrm{CO}_2(aq) + \mathrm{H}_2\mathrm{O}(l) \longleftrightarrow \mathrm{H}_2\mathrm{CO}_3(aq)\]
While Henry's Law gives us the concentration of dissolved \(\mathrm{CO}_2\), it is the chemical equilibrium that helps us to understand the distribution between \(\mathrm{CO}_2\) and \(\mathrm{H}_2\mathrm{CO}_3\) in solution, impacting the overall system's behavior, including pH values if \(\mathrm{H}_2\mathrm{CO}_3\) were to further dissociate into ions.

pH Calculation from Gas Solubility

Calculating the pH of a solution involves understanding the concentration of hydrogen ions (\(\mathrm{H}^+\)) present in that solution. pH is a measure of the acidity or alkalinity of an environment and is calculated using the following logarithmic scale:
\[pH = -\log_{10}[\mathrm{H}^+]\]
When \(\mathrm{CO}_2\) is dissolved in water, it forms carbonic acid (\(\mathrm{H}_2\mathrm{CO}_3\)), which can dissociate to produce hydrogen ions. Though \(\mathrm{H}_2\mathrm{CO}_3\) is a weak acid, for simplicity, one may assume that the concentration of \(\mathrm{H}^+\) is proportional to the concentration of \(\mathrm{H}_2\mathrm{CO}_3\). This assumption simplifies the calculation of pH, allowing us to understand the acid-base characteristics of solutions formed by dissolved gases. However, it's worth noting that for more accurate pH calculations, one should consider the acid's dissociation constant (\(K_a\)) and degree of dissociation in the solution.