Question

The ratio of dissociation constant of two weak acids \(\mathrm{Hx}\) and \(\mathrm{Hy}\) is 4 . At what molar concentration ratio, the two acids can have same \(\mathrm{pH}\) value? (a) 1 (b) \(0.75\) (c) \(0.25\) (d) 4

Short answer

The concentration ratio for the two acids to have the same pH is 0.25.

Step-by-step solution

Understanding the problem

We need to determine at what concentration ratio the two weak acids \(\mathrm{Hx}\) and \(\mathrm{Hy}\), with a dissociation constant ratio of 4, have the same pH.

Recall the relationship of pH and dissociation constant

For weak acids, the pH can be related to the dissociation constant \(K_a\) and concentration \([A]\) using the formula: \[ \text{pH} = -\log\sqrt{K_a \cdot [A]} \].

Express the condition for same pH

For \(\mathrm{Hx}\) and \(\mathrm{Hy}\) to have the same pH, their formula \( -\log\sqrt{K_{a_{Hx}} \cdot [Hx]} = -\log\sqrt{K_{a_{Hy}} \cdot [Hy]} \) must be equal.

Substitute the ratio of dissociation constants

With \(K_{a_{Hx}} / K_{a_{Hy}} = 4\), let's set \(K_{a_{Hx}} = 4K_{a_{Hy}}\). Substitute these into the pH equality: \[ \sqrt{4K_{a_{Hy}} \cdot [Hx]} = \sqrt{K_{a_{Hy}} \cdot [Hy]} \].

Simplify the equation

Divide and simplify both sides: \[ \sqrt{4} \times \sqrt{K_{a_{Hy}}} \times \sqrt{[Hx]} = \sqrt{K_{a_{Hy}}} \times \sqrt{[Hy]} \] leading to \(2\sqrt{[Hx]} = \sqrt{[Hy]}\).

Square to eliminate square roots

Square both sides to eliminate the square roots: \[ 4[Hx] = [Hy] \].

Calculate the concentration ratio

The concentration ratio of \([Hx] / [Hy] = 1/4 = 0.25\). Thus, option (c) is correct.

Key concepts

Dissociation Constant

The dissociation constant, often represented as \( K_a \), is a critical parameter in understanding the strength of weak acids. It indicates how readily an acid dissociates into its ions in a solution. For weak acids, \( K_a \) is relatively small due to their partial ionization.
To determine \( K_a \) for any acid, you can use the equilibrium expression: \[ K_a = \frac{[H^+][A^-]}{[HA]} \]
This formula highlights the concentration of the hydrogen ions \([H^+]\) and the anion \( [A^-] \) in comparison to the non-ionized acid \([HA]\).

• High \( K_a \) indicates a stronger tendency to dissociate.
• Low \( K_a \) suggests limited ionization.

The dissociation constant is pivotal when comparing the acidity of two weak acids, as it directly influences their pH behavior.

Molar Concentration

Molar concentration, often referred to as molarity, is the number of moles of a solute present in a one-liter solution. It is a fundamental measure in chemistry that determines the concentration of solutions in various chemical reactions.
Represented as \([A]\) for a solute \(A\), it is calculated using the formula: \[ [A] = \frac{moles \: of \: solute}{volume \: in \: liters} \]
In the context of weak acids, the molar concentration of the acid itself is pivotal in predicting the extent of its dissociation. When discussing weak acids \( ext{Hx} \) and \( ext{Hy} \), knowing their molar concentrations helps characterize their pH values and guides chemical calculations.

• Molar concentration links directly to the hydroxide and hydrogen ion concentrations.
• It aids in calculating the dissociation constant \( K_a \), which is crucial for determining acid strength.

Thus, understanding molar concentration helps in assessing how acids influence their environment and react within solutions.

pH Calculation

pH is a scale used to specify the acidity or basicity of an aqueous solution. For weak acids, calculating the pH involves understanding the relationship between the dissociation constant and its concentration.
The general formula for pH calculation in context with weak acids is given by: \[ \text{pH} = -\log\sqrt{K_a \cdot [A]} \]
Here's how it works:

• Combine the known values of \( K_a \) and \([A]\) to approximate the concentration of hydrogen ions.
• Use the logarithmic function to translate the hydrogen ion concentration to the pH value.

This approach simplifies determining how acidic a solution of a weak acid is, by leveraging relatively accessible calculations.

Concentration Ratio

The concentration ratio is a comparative measure used to establish relationships between different solute concentrations. In the example of weak acids \( ext{Hx} \) and \( ext{Hy} \), determining their concentration ratio is vital to finding conditions where both have the same pH.
Given two weak acids with differing dissociation constants, the concentration ratio helps balance these differences for equal pH levels. It is often essential to exploit the relationship between the dissociation constants and concentrations.
To find the concentration ratio at which two acids have the same pH, we relate their dissociation equations such that:

• \( 2\sqrt{[Hx]} = \sqrt{[Hy]} \)
• Squaring both sides offers a straightforward solution.

The derived equation \( 4[Hx] = [Hy] \) reveals that the concentration ratio for \( [Hx]/[Hy] \) needed for the same pH is \( 0.25 \). By calculating this, we can directly compare and adjust the concentrations to match desired pH levels, essential for precise experimental conditions.