The concept used to solve this question is to match the ${\rm{pH}}$ with the corresponding buffer solutions A, B and C.
$pH$ is the measure of hydrogen ion concentration. Lower the $pH$ , higher is the hydrogen ion concentration and lower is the hydroxide ion concentration.
An equilibrium constant for the dissociation reaction is the equation expressing the extent of dissociation into ions which is equal to the product of the concentrations of the respective ions divided by the concentration of the undissociated molecule.
Consider a dissociation reaction of acetic acid, ${\rm{C}}{{\rm{H}}_3}{\rm{COOH}} + {{\rm{H}}_2}{\rm{O}}\longrightarrow{{}}{{\rm{H}}_3}{{\rm{O}}^ + } + {\rm{C}}{{\rm{H}}_3}{\rm{CO}}{{\rm{O}}^ - }$ .
The equilibrium constant for the dissociation of ${\rm{HA}}$ is given as follows:
${K_a} = \frac{{\left[ {{{\rm{H}}_3}{{\rm{O}}^ + }} \right]\left[ {{\rm{C}}{{\rm{H}}_3}{\rm{CO}}{{\rm{O}}^ - }} \right]}}{{\left[ {{\rm{C}}{{\rm{H}}_3}{\rm{COOH}}} \right]}}$
$p{K_a}$ is also the measure of acidic strength.
The formula relating ${K_a}$ and $p{K_a}$ is as follows:
$p{K_a} = - \log {K_a}$
The Henderson-Hasselbalch equation to calculate the ${\rm{pH}}$ of a buffer solution is as follows:
${\rm{pH}} = p{K_a} + \log \frac{{\left[ {{\rm{C}}{{\rm{H}}_3}{\rm{CO}}{{\rm{O}}^ - }} \right]}}{{\left[ {{\rm{C}}{{\rm{H}}_3}{\rm{COOH}}} \right]}}$
Given, ${K_a} = 1.8 \times {10^{ - 5}}$
The $p{K_a}$ of the acid is calculated as follows:
$\begin{array}{l}\\p{K_a} = - \log \left( {1.8 \times {{10}^{ - 5}}} \right)\\\\{\rm{ = 4}}{\rm{.74}}\\\end{array}$
Given that, for buffer A,
$\left[ {{\rm{Acetic acid}}} \right]$ is 10 times greater the concentration of $\left[ {{\rm{Acetate}}} \right]$ .
So,
$\begin{array}{l}\\\left[ {{\rm{Acetic acid}}} \right] = 10x\\\\\left[ {{\rm{Acetate}}} \right] = x\\\end{array}$
Substitute the values in the Henderson-Hasselbalch equation and calculate the ${\rm{pH}}$ of the buffer as follows:
$\begin{array}{l}\\{\rm{pH}} = 4.74 + \log \frac{x}{{{\rm{10}}x}}\\\\{\rm{ = 3}}{\rm{.74}}\\\end{array}$
Given that, for buffer B,
$\left[ {{\rm{Acetate}}} \right]$ is 10 times greater the concentration of $\left[ {{\rm{Acetic acid}}} \right]$ .
So,
$\begin{array}{l}\\\left[ {{\rm{Acetic acid}}} \right] = x\\\\\left[ {{\rm{Acetate}}} \right] = 10x\\\end{array}$
Substitute the values in the Henderson-Hasselbalch equation and calculate the ${\rm{pH}}$ of the buffer as follows:
$\begin{array}{l}\\{\rm{pH}} = 4.74 + \log \frac{{10x}}{x}\\\\{\rm{ = 5}}{\rm{.74}}\\\end{array}$
Given that, for buffer C,
$\left[ {{\rm{Acetic acid}}} \right] = \left[ {{\rm{Acetate}}} \right]$ .
So,
$\begin{array}{l}\\\left[ {{\rm{Acetic acid}}} \right] = x\\\\\left[ {{\rm{Acetate}}} \right] = x\\\end{array}$
Substitute the values in the Henderson-Hasselbalch equation and calculate the ${\rm{pH}}$ of the buffer as follows:
$\begin{array}{l}\\{\rm{pH}} = 4.74 + \log \frac{x}{x}\\\\{\rm{ = 4}}{\rm{.74}}\\\end{array}$
Ans:The ${\rm{pH}}$ of the three buffer solutions are as follows:
$\begin{array}{l}\\{\rm{Buffer A}}:3.74\\\\{\rm{Buffer B}}:5.74\\\\{\rm{Buffer C}}:4.74\\\end{array}$
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