Question

When calculating \(\left[\mathrm{H}_{3} \mathrm{O}^{+}\right]\) for weak acid solutions, we can often use the \(x\) is small approximation. Explain the nature of this approximation and why it is valid.

Short answer

The 'x is small' approximation assumes that the change in acid concentration due to dissociation is negligible when compared to the initial concentration, and is valid when the acid dissociation constant is much smaller than the initial concentration. It is often used when the dissociation percentage is under 5%.

Step-by-step solution

Understand the Approximation

The 'x is small' approximation is used in weak acid equilibrium calculations where the acid does not dissociate completely. In the acid dissociation equation, HA ⇌ H₃O⁺ + A⁻, 'x' represents the concentration of H₃O⁺ and A⁻ formed in the dissociation. The approximation assumes that the amount of HA that dissociates is much less than the initial concentration of HA, so the change in concentration of HA (which is 'x') can be considered small enough to be negligible.

Justify the Validity of the Approximation

This approximation is valid when the acid dissociation constant (Ka) is much smaller than the initial concentration of the acid. Mathematically, if the initial concentration of the acid is 'a' and 'x' is the change, then for Ka<<a, (a - x) ≈ a since 'x' is very small in comparison to 'a'. Under these conditions, the change in concentration does not significantly impact the overall concentration of the acid, allowing for a simplification of the calculations.

Determine When to Use the Approximation

The approximation is typically valid when the dissociation percentage is less than 5% of the initial concentration. To check if the approximation is reasonable, calculate the percent ionization after assuming the approximation. If the percent ionization is indeed less than 5%, then the approximation is valid. Otherwise, a more exact method, like solving the quadratic equation for 'x', will be necessary.

Key concepts

Acid Dissociation Constant

The acid dissociation constant (Ka) is a quantitative measure of how completely an acid dissociates in water. It is represented by the equation \(Ka = \frac{[H_3O^+][A^-]}{[HA]}\), where [HA] is the concentration of the undissociated acid, and [H3O+] and [A-] are the concentrations of the hydronium and conjugate base formed from the dissociation of the acid, respectively.

In simple terms, Ka tells us the strength of an acid: the larger the Ka, the stronger the acid and the more it dissociates. For weak acids, which only partly dissociate, Ka is usually very small, which is critical for making the 'x is small' approximation.

Percent Ionization

When discussing weak acids, the percent ionization, also known as percent dissociation, is an important concept. It represents the fraction of acid molecules that have dissociated into ions, expressed as a percent. You can calculate it using the equation \(\text{Percent Ionization} = \frac{[H_3O^+]}{[HA]_{initial}} \times 100\%\).

The smaller the percent ionization, the weaker the acid. If the percent ionization is found to be less than 5%, the 'x is small' approximation can be considered reliable, and conversely, it helps to assess the validity of the approximation in a given weak acid equilibrium scenario.

Equilibrium Calculation

An equilibrium calculation is used to find the concentrations of reactants and products in a chemical reaction that has reached equilibrium. For weak acids, this can often be done using equilibrium constants like Ka. The key is to set up an 'ICE' table (Initial, Change, Equilibrium) to keep track of concentration changes. However, if we make the 'x is small' approximation, the equilibrium calculation becomes simpler as we assume that the initial concentration of the acid hardly changes, which means \( [HA]_{equilibrium} \approx [HA]_{initial} \), simplifying the equation needed to find the concentration of ions.

Concentration Approximation

The concentration approximation technique simplifies calculations for weak acid equilibria by assuming the change in concentration of the acid (denoted by 'x') due to its dissociation is much less than the initial concentration. Hence, mathematically, we have \( [HA]_{initial} - x \approx [HA]_{initial} \).

This approximation is particularly useful for weak acids with a low acid dissociation constant. However, this approximation only holds true when the percent ionization is small (typically less than 5%), meaning that it only applies to very weakly dissociating systems. If the 'x is small' approximation does not hold, a more precise method must be employed, such as a quadratic formula, to adequately solve for the concentrations at equilibrium.