Question

When determining the pH of a weak acid solution, sometimes the 5$\mathrm{\%}$ rule can be applied to simplify the math. At what $K_{\mathrm{a}}$ values will a $1.0-M$ solution of a weak acid follow the 5$\mathrm{\%}$ rule?

Short answer

For a 1.0 M solution of a weak acid, the 5% rule can be applied when $K_{\mathrm{a}}$ values are less than or equal to 0.0025.

Step-by-step solution

Understand the 5% rule

The 5% rule is used to simplify the calculation of the equilibrium concentration of a weak acid and its conjugate base in an aqueous solution. According to the 5% rule, if the dissociation of a weak acid is less than 5% of the initial concentration, then the change in concentration due to dissociation can be neglected. Mathematically, this can be expressed as: $x\le 0.05\ast C$ where x represents the change in concentration due to dissociation and C represents the initial concentration of the weak acid.

Relate the 5% rule to the Ka expression

For a weak acid, HA, the equilibrium equation in an aqueous solution can be represented as: $HA<=>H^{+}+A^{-}$ The equilibrium constant, $K_{\mathrm{a}}$, for this reaction is given by: $K_{\mathrm{a}}=\frac{[H^{+}][A^{-}]}{[HA]}$ Let's assume that the change in concentration due to dissociation is x. The concentration of each species at equilibrium can then be represented as: $[H^{+}]=[A^{-}]=x$ $[HA]=C-x$ Substitute these values into the $K_{\mathrm{a}}$ expression: $K_{\mathrm{a}}=\frac{x^2}{C-x}$ Using the 5% rule, we can assume that x << C, so C - x ≈ C. Then, the simplified expression for $K_{\mathrm{a}}$ is: $K_{\mathrm{a}}\approx \frac{x^2}{C}$ Now, we need to find the range of $K_{\mathrm{a}}$ values that will satisfy this approximation.

Determine the Ka values that satisfy the 5% rule

From Step 1, we know that x ≤ 0.05 * C. Since the initial concentration of the weak acid, C, is given as 1.0 M, we can now write: $x\le 0.05\ast 1.0=0.05$ Now, we substitute this into the simplified $K_{\mathrm{a}}$ expression from Step 2: $K_{\mathrm{a}}\approx \frac{x^2}{C}\le \frac{(0.05{)}^2}{1.0}$ Calculate this inequality: $K_{\mathrm{a}}\le 0.0025$ So, for a 1.0 M solution of a weak acid, the 5% rule can be applied when $K_{\mathrm{a}}$ values are less than or equal to 0.0025.

Key concepts

Weak Acid Equilibrium

When dealing with weak acids, it's important to understand the nature of weak acid equilibrium. Weak acids partially dissociate in water, meaning they only release a small fraction of their hydrogen ions into the solution. This is why the equilibrium state is crucial, as it represents the balance between the undissociated acid and its dissociated ions.

In the equilibrium reaction for a weak acid (HA), it can be represented by the equation:

• $HA\leftrightarrow H^{+}+A^{-}$

This implies that at equilibrium, the concentration of the hydrogen ions $[H^{+}]$ and the conjugate base $[A^{-}]$ remains constant opposed to any further change. In weak acid solutions, the equilibrium lies mainly towards the undissociated acid (HA) as opposed to the ions.

This equilibrium is dynamic but governed by the reaction's acid dissociation constant $K_a$. However, the key takeaway is that because a weak acid only partially dissociates, we have to take equilibrium into account to accurately describe its properties.

Acid Dissociation Constant (Ka)

The acid dissociation constant, represented as $K_a$, is a vital concept in understanding weak acids. It's a measure of the strength of an acid in solution; specifically, how well an acid dissociates into its ions.

Mathematically, $K_a$ is defined by the equation:

• $K_a=\frac{[H^{+}][A^{-}]}{[HA]}$

Here, $[H^{+}]$ and $[A^{-}]$ are the concentrations of the dissociated hydrogen ions and conjugate base, and $[HA]$ is the concentration of the undissociated acid at equilibrium. A large $K_a$ value indicates a strong acid, which means more of the acid dissociates into ions. Conversely, a small $K_a$ value signifies a weak acid, as it shows limited dissociation.

The $K_a$ value is crucial for determining the pH and behavior of acid solutions. It allows chemists to predict how an acid will react in different environments and is fundamental in applications such as calculating pH for titrations or understanding buffer solutions.

pH Calculation of Weak Acids

Calculating the pH of weak acids can sometimes be simplified using the 5% rule, especially when dealing with very weak acids. The pH is a measure of hydrogen ion concentration in the solution, but for weak acids, we often have to solve for an equilibrium concentration.

To find the pH, you need to determine the concentration of $[H^{+}]$ ions at equilibrium, which involves using the $K_a$ value of the acid. For a weak acid with equilibrium reaction:

• $HA\leftrightarrow H^{+}+A^{-}$

The initial concentration of $HA$ is represented as $C$, and the change in concentration due to dissociation is $x$, leading to:

• $[H^{+}]=x$
• $[HA]=C-x$

Using the formula $K_a=\frac{x^2}{C-x}$, and assuming $x<<C$, means $C-x\approx C$, simplifying calculations to:

• $K_a\approx \frac{x^2}{C}$

From here, solving for $x$ gives us the $[H^{+}])$ concentration, and the pH can be calculated as $pH=-\log [H^{+}]$.

The 5% rule applies when $x\le 0.05C$, ensuring that the simplification is valid and the pH calculated is accurate.