Question

A \(1.0 \times 10^{-2}-M\) solution of cyanic acid (HOCN) is 17\(\%\) dissociated. Calculate \(K_{\mathrm{a}}\) for cyanic acid.

Short answer

The Ka for cyanic acid can be calculated using the equilibrium concentrations of the dissociation products and the undissociated acid: \(K_a = \frac{(0.0017)(0.0017)}{0.0083} \approx 3.48 \times 10^{-4}\).

Step-by-step solution

Write the dissociation equation for the acid

Write the dissociation equation for cyanic acid (HOCN) in water: \(HOCN \longleftrightarrow H^+ + OCN^-\)

Determine the initial concentrations of HOCN, H+, and OCN-

We are given a 0.01 M solution of HOCN. Both the H+ and OCN- ions will be formed as the dissociation progresses, so their initial concentrations will be 0. Initial concentrations: [HOCN] = 0.01 M [H+] = 0 M [OCN-] = 0 M

Determine the equilibrium concentrations of HOCN, H+, and OCN-

Since 17% of the HOCN molecules have dissociated, we can determine the equilibrium concentrations by considering a decrease of 0.17 * 0.01 M for HOCN and an increase of the same amount for H+ and OCN-: Equilibrium concentrations: [HOCN] = 0.01 M - 0.17 * 0.01 M [H+] = 0 + 0.17 * 0.01 M [OCN-] = 0 + 0.17 * 0.01 M Calculate the values: [HOCN] = 0.01 M - (0.17 * 0.01 M) = 0.0083 M [H+] = 0.17 * 0.01 M = 0.0017 M [OCN-] = 0.17 * 0.01 M = 0.0017 M

Write the equilibrium constant expression (Ka) and calculate Ka

Now that we have the equilibrium concentrations of all species, we can write the expression for Ka: \(K_a = \frac{[H^+][OCN^-]}{[HOCN]}\) Plug in the equilibrium concentrations: \(K_a = \frac{(0.0017)(0.0017)}{0.0083}\) Calculate Ka: \(K_a \approx 3.48 \times 10^{-4}\) So the Ka for cyanic acid is approximately \(3.48 \times 10^{-4}\).

Key concepts

Cyanic Acid

Cyanic acid, with the chemical formula HOCN, is a weak acid. In chemistry, acids are substances that can donate a proton (H⁺ ion) in a reaction. Cyanic acid partially dissociates in water, meaning that not all of its molecules break down into ions. This partial dissociation is typical of weak acids.
In a solution, cyanic acid dissociates into H⁺ and OCN⁻ ions. The extent of dissociation helps us understand how the acid behaves in solution and also helps calculate important chemical constants like the acid dissociation constant, or Kₐ.

Equilibrium Concentrations

When cyanic acid is dissolved in water, it establishes an equilibrium between the undissociated HOCN molecules and the ions produced, H⁺ and OCN⁻.
Equilibrium concentrations refer to the concentrations of all species in a reaction when it has reached a state of balance.

• Initially, you start with a known concentration of HOCN, for example, 0.01 M.
• No ions are present initially, so their concentrations begin at 0 M.
• As 17% of HOCN dissociates, we calculate the change in concentration for each species in the reaction.

This understanding is crucial because, at equilibrium, the rate at which HOCN dissociates into ions is equal to the rate at which the ions recombine to form HOCN molecules. Hence, the concentrations remain constant over time.

Dissociation Equation

The dissociation equation for cyanic acid is a chemical equation that describes how the acid breaks into ions. It is written as:
\[ HOCN \longleftrightarrow H^+ + OCN^- \]
This equation illustrates the process in which cyanic acid (HOCN) separates into hydrogen ions (H⁺) and cyanate ions (OCN⁻) in water.
Analyzing the dissociation helps us understand the amount of acid that ionizes and how the different species are distributed at equilibrium. This equation is vital in calculating equilibrium concentrations and for understanding the reaction dynamics of weak acids like cyanic acid.

Equilibrium Constant Expression

The equilibrium constant expression for a dissociation reaction is a crucial concept in chemistry.
It describes the ratio of the concentrations of the products to the reactants, each raised to the power of their coefficients in the balanced dissociation equation.
For cyanic acid, the equilibrium constant expression (Kₐ) is:
\[ K_a = \frac{[H^+][OCN^-]}{[HOCN]} \]
This expression allows chemists to calculate Kₐ, the acid dissociation constant, which reflects the strength of the acid. The larger the Kₐ, the stronger the acid and the more it dissociates in solution. In this exercise, given the equilibrium concentrations, we calculated that:
\[ K_a \approx 3.48 \times 10^{-4} \]
Interpreting this value helps in understanding how cyanic acid behaves in a solution and is essential for predicting the outcome of reactions involving this acid.