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Chapter 15: Problem 138

A \(0.400 M\) formic acid (HCOOH) solution freezes at \(-0.758^{\circ} \mathrm{C}\). Calculate the \(K_{\mathrm{a}}\) of the acid at that temperature. (Hint: Assume that molarity is equal to molality. Carry your calculations to three significant figures and round off to two for \(K_{\mathrm{a}}\).)

Short Answer

Expert verified
Compute the steps to find the accurate freezing point depression and the degree of ionization (\(\alpha\)). Using the degree of ionization, solve the \(K_a\) expression with the given molarity. The final \(K_a\) value, rounded off to two significant figures, is the desired answer.

Step by step solution

01

Determine and write the formula for the freezing point depression

The freezing point depression for a solution is given by \(\Delta T_f = K_f \cdot m \cdot i\), where \(\Delta T_f\) is the change in freezing temperature, \(K_f\) is the cryoscopic constant of water (\(1.86^{\circ}\mathrm{C/m}\)), \(m\) is the molality of the solution (which is taken as 0.400 M in this question) and \(i\) is the van 't Hoff factor (which relates to the degree of dissociation of the solute). For formic acid, which is a weak acid and only partially ionises, there is a need first to find \(i\), which we denote as \(i = 1 + \alpha\) where \(\alpha\) is the degree of ionization.
02

Substitute the known values and rearrange the formula for the degree of ionization

We are given the \(\Delta T_f\) as \(0.758^{\circ}\mathrm{C}\), so substitute this value in, and then rearrange the formula to solve for \(i\). The rearranged formula is \(i = \frac{\Delta T_f}{K_f \cdot m}\). When substituted, the equation becomes \(i = \frac{0.758^{\circ}\mathrm{C}}{(1.86^{\circ}\mathrm{C/m) \cdot (0.400 M)} = 1 + \alpha\). This will give the value of \(i\), which we can then use to find \(\alpha\).
03

Calculate the degree of ionization

On calculation from the last equation, \(i = 1 + \alpha\) gives \(\alpha = i - 1\). This value represents the degree of ionization, and will be used to find \(K_a\) in the next step.
04

Setup and solve the \(K_a\) Expression

The equilibrium for the dissolution and ionization of formic acid is given as: \(HCOOH \leftrightarrow H^+ + HCOO^-\). The \(K_a\) expression related to this reaction is \(K_a = \frac{[H^+][HCOO^-]}{[HCOOH]}\). Since every mol of formic acid dissociates to produce H+ and HCOO-, and we are given that \(i = \alpha\) (since this is a weak acid) the concentrations for each of these is \(\alpha[M]\), and the leftover [HCOOH] is \((1-\alpha)[M]\). Substituting these values, the \(K_a\) equation becomes: \(K_a = \frac{\alpha^2 \cdot M}{(1-\alpha) \cdot M}\). Solving this equation with \(\alpha\) value from step 3 and \(M = 0.400M\), gives us the \(K_a\) value of the formic acid.
05

Round off the \(K_a\) value

The \(K_a\) value calculated from step 4 needs to be rounded off to two significant figures as per the hint in the given exercise. This will provide the accurate \(K_a\) value for the formic acid at the given temperature.

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Most popular questions from this chapter

Consider the two weak acids HX (molar mass = \(180 \mathrm{~g} / \mathrm{mol}\) ) and \(\mathrm{HY}\) (molar mass \(=78.0 \mathrm{~g} / \mathrm{mol}\) ). If a solution of \(16.9 \mathrm{~g} / \mathrm{L}\) of \(\mathrm{HX}\) has the same \(\mathrm{pH}\) as one containing \(9.05 \mathrm{~g} / \mathrm{L}\) of \(\mathrm{HY},\) which is the stronger acid at these concentrations?

Give an example of the following: (a) a weak acid that contains oxygen atoms, (b) a weak acid that does not contain oxygen atoms, (c) a neutral molecule that acts as a Lewis acid, (d) a neutral molecule that acts as a Lewis base, (e) a weak acid that contains two ionizable \(\mathrm{H}\) atoms, (f) a conjugate acidbase pair, both of which react with \(\mathrm{HCl}\) to give carbon dioxide gas.

Both the amide ion \(\left(\mathrm{NH}_{2}^{-}\right)\) and the nitride ion \(\left(\mathrm{N}^{3-}\right)\) are stronger bases than the hydroxide ion and hence do not exist in aqueous solutions. (a) Write equations showing the reactions of these ions with water, and identify the Brønsted acid and base in each case. (b) Which of the two is the stronger base?

Like water, liquid ammonia undergoes autoionization: $$ \mathrm{NH}_{3}+\mathrm{NH}_{3} \rightleftharpoons \mathrm{NH}_{4}^{+}+\mathrm{NH}_{2}^{-} $$ (a) Identify the Brønsted acids and Brønsted bases in this reaction. (b) What species correspond to \(\mathrm{H}^{+}\) and \(\mathrm{OH}^{-}\) and what is the condition for a neutral solution?

Without referring to the text, write the formulas of four strong acids and four weak acids.
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