Question

The ionization constant of acetic acid is $1.74\times {10}^{-5}$. Calculate the degree of dissociation of acetic acid in its 0.05 M solution. Calculate the concentration of acetate ion in the solution and its pH.

Answer 1

Method 1
1) $C{H}_{3}COOH\leftrightarrow C{H}_{3}CO{O}^{-}+H^{+}{K}_a=1.74\times {10}^{-5}$
2) $H_{2}O+H_{2}O\leftrightarrow H_3{O}^{+}+O{H}^{-}{K}_w=1.0\times {10}^{-14}$
Since $Ka>>K_2$:
$\begin{array}{cccccc}& C{H}_{3}COOH+H_{2}O& \leftrightarrow & C{H}_{C}O{O}^{-}& +& H_3{O}^{+}\\ c_1=& 0.05& & 0& & 0\\ & 0.05-.05\alpha & & 0.05\alpha & & 0.05\alpha \end{array}$
$K_a=\frac{\left(.05\alpha \right)\left(.05\alpha \right)}{(.05\alpha -0.05\alpha )}=\frac{\left(.05\alpha \right)\left(0.05\alpha \right)}{.05(1-\alpha )}=\frac{.05{\alpha }^2}{1-\alpha }1.74\times {10}^{-5}=\frac{0.05{\alpha }^2}{1-\alpha }1.74\times {10}^{-5}-1.74\times {10}^{-5}\alpha =0.05{\alpha }^{2}0.05{\alpha }^2+1.74\times {10}^{-5}\alpha -1.74\times {10}^{-5}D=b^2-4ac=(1.74\times {10}^{-5}{)}^2-4(.05\left)\right(1.74\times {10}^{-5})=3.02\times {10}^{-25}+.348\times {10}^{-5}\alpha =\sqrt{\frac{K_a}{c}}\alpha =\sqrt{\frac{1.74\times {10}^{-5}}{.05}}=\sqrt{\frac{34.8\times {10}^{-5}\times 10}{10}}=\sqrt{3.48\times {10}^{-6}}=C{H}_{3}COOH\leftrightarrow C{H}_{3}CO{O}^{-}+H^{+}\frac{\alpha 1.86\times {10}^{-3}}{\left[C{H}_{3}CO{O}^{-}\right]}=0.05\times 1.86\times {10}^{-3}=\frac{0.93\times {10}^{-3}}{1000}=.000093$

Method 2
Degree of dissociation,
$\alpha =\sqrt{\frac{K_a}{c}}C=0.05M{K}_a=1.74\times {10}^{-5}$
Then, $\alpha =\sqrt{\frac{1.74\times {10}^{-5}}{.05}}\alpha =\sqrt{34.8\times {10}^{-5}}\alpha =\sqrt{3.48}\times {10}^{-4}\alpha ={1.8610}^{-2}C{H}_{3}COOH\leftrightarrow C{H}_{3}COOH\leftrightarrow C{H}_{3}CO{O}^{-}+H^{+}$
Thus, concentration of $C{H}_{3}CO{O}^{-}=C.a$
$=0.5\times 1.86\times {10}^{-2}=.093\times {10}^{-2}=.00093M$
Since $\left[oA{c}^{-}\right]=\left[H^{+}\right]\left[H^{+}\right]=.00093=.093\times {10}^{-2}pH=-log\left[H^{+}\right]=-log(.093\times {10}^{-2})\therefore pH=3.03$
Hence, the concentration of acetate ion in the solution is 0.00093 M and its Ph is 3.03.