Question

Student Name: Instructor Name: 3. A poorly calibrated pH meter yields a pH of 3.70 when the true value of the pH is 4.00. If these measurements were taken with the intent of calculating the K, of a 0.10 M solutio of a weak acid, calculate The % error of the pH The % error of the [H3O*] .The % error of K Show ALL work in your calculations. 13.70-4.00 -17.5 * (०০ 4.00

Answer 1

a)

True value of pH= 4.00

pH meter reading = 3,70

$\%\;error=\frac{\textrm{pH Meter value - True Value}}{\textrm{True Value}}\times100\\ \Rightarrow \%\;error=\frac{{3.70 - 4.00}}{4.00}\times100\\ \Rightarrow \%\;error=-7.5\%$

Thus, % error in pH is -7.5%

b)

We know that the concentration of Hydronium ion is related to pH as,

$pH=-\log_{10}{[H_3O^+]}\\ \Rightarrow [H_3O^+]=10^{-pH}$

Actual concentration,

$[H_3O^+]_{True}=10^{-4.00}$

pH meter concentration,

$[H_3O^+]_{Measured}=10^{-3.70}$

Percentage error,

$\%\;error=\frac{[H_3O^+]_{Measured}-[H_3O^+]_{True}}{[H_3O^+]_{True}}\times100\\ \Rightarrow \%\;error=\frac{10^{-3.70} - 10^{-4.00}}{10^{-4.00}}\times100\\ \Rightarrow \%\;error=99.5\%$

Thus, the % error in [H₃O⁺] is 99.5%

c)

Acetic Acid is a weak acid. Thus, the acid dissociation constant $\alpha$ is given as,

$\alpha=\frac{[H_3O^+]}{c}$

The K_a is given as,

$K_a=\frac{c\alpha^2}{1-\alpha}\\ \Rightarrow K_a=\frac{c\left ( \frac{[H_3O^+]}{c} \right )^2}{1-\left( \frac{[H_3O^+]}{c} \right )}\\ \Rightarrow K_a =\frac{[H_3O^+]^2}{c-[H_3O^+]}$

For actual K_a

$(K_a)_{True} =\frac{[H_3O^+]_{True}^2}{c-[H_3O^+]_{True}}\\ \Rightarrow (K_a)_{True} =\frac{(10^{-4.00})^2}{0.10-10^{-4.00}}\\ \Rightarrow (K_a)_{True}\approx 1.00\times10^{-7}$

For measured reading,

$(K_a)_{Measured} =\frac{[H_3O^+]_{Measured}^2}{c-[H_3O^+]_{Measured}}\\ \Rightarrow (K_a)_{Measured} =\frac{(10^{-3.70})^2}{0.10-10^{-3.70}}\\ \Rightarrow (K_a)_{Measured}\approx 3.99\times 10^{-7}$

% Error in K_a

$\%\;error=\frac{(K_a)_{Measured}-(K_a)_{True}}{(K_a)_{True}}\times100\\ \Rightarrow \%\;error=\frac{3.99\times10^{-7} - 1.00\times10^{-7}}{1.00\times10^{-7}}\times100\\ \Rightarrow \%\;error\approx299\%$

Thus, we see that a pH error of -7.5% can translate into error in measuring [H₃O⁺] by approx 99.5% and that of K_a by approx 299%