Question

The $pH$ of a solution is $10$ and that of another is $12$. When equal volumes of these two are mixed, the $pH$ of the resulting solution is :
A.) $10$
B.) $12$
C.) $11$
D.) $10.3010$

Answer 1

Hint: This question can be solved by the concept that when equal volumes of two solutions are mixed together then the concentration of hydrogen ion that is $[{H^ + }]$ of each solution will get half. Then the sum of these concentrations gives us total concentration of the resulting solution.

Complete step by step answer:
In this question, the $pH$ value of the solution can be defined as the negative of the logarithm of the hydrogen ion. It is the measure of the acidity of the solution or the alkalinity of the solution. If the $pH$ of the solution is less than seven then it is an acidic solution, if it is equal to seven then it is a neutral solution and if $pH$ of the solution is more than seven then it is a basic solution. The $pH$ of the solution can be written as:
$pH = - \log [{H^ + }]$
Or we can also write it as,
$[{H^ + }] = {10^{ - pH}}$
Now, for the first solution we have given the $pH = 10$, therefore the value of concentration of hydrogen ion for solution $ - 1$ is $[{H^ + }] = {10^{ - 10}}$
Also, for the second solution we have given the $pH = 12$, therefore the value of concentration of hydrogen ion for solution $ - 2$ is $[{H^ + }] = {10^{ - 12}}$
As we know that when the equal volume of two solutions are mixed together then the concentration of $[{H^ + }]$ for each solution gets half. Therefore, the concentration of hydrogen ion for final solution is given as:
$
  [{H^ + }] = \dfrac{{{{[{H^ + }]}_1}}}{2} + \dfrac{{{{[{H^ + }]}_2}}}{2} \\
  [{H^ + }] = \dfrac{{{{10}^{ - 10}}}}{2} + \dfrac{{{{10}^{ - 12}}}}{2} \\
  [{H^ + }] = {10^{ - 10}} \times (0.5 + 0.005) \\
  [{H^ + }] = 0.505 \times {10^{ - 10}} \\
 $
Now, $pH$ of the resulting solution is :
$
  pH = - \log [0.505 \times {10^{ - 10}}] \\
  pH = - \{ \log [0.505] + \log [{10^{ - 10}}]\} \\
  pH = - \{ - 0.30 - 10\} \\
  pH = 10.30 \\
 $
As the required $pH$ value of the resultant solution is approximately $10.30$.

Hence, option D.) is the correct answer.

Note:
Always remember that when equal volumes of two solutions are mixed together then the resultant is not the average of the $pH$ values of both solutions but rather it is the average of the concentration of hydrogen ion that is given as $[{H^ + }]$.