Explain the $pH$ scale and $pH$.

(N/A) $pH$ Scale: The hydronium ion concentration in molarity is more conveniently expressed on a logarithmic scale known as the $pH$ scale.
Definition of $pH$: The $pH$ of a solution is defined as the negative logarithm of the activity of hydrogen ions $(a_{H^{+}})$.
Since activity $a$ is dimensionless,$a_{H^{+}} = [H^{+}] / (1 \ mol \ L^{-1})$.
This definition is expressed as:
$pH = -\log a_{H^{+}} \quad \dots (Eq.-I)$
For dilute solutions where the concentration is expressed in $mol \ L^{-1}$:
$pH = -\log [H^{+}] \quad \dots (Eq.-II)$
The $pH$ scale is suitable for dilute solutions where $[H^{+}] < 1 \ M$.
Temperature dependence: The variation of $pH$ with temperature is generally small and often ignored.
Logarithmic nature: Since the $pH$ scale is logarithmic,a change of $1$ unit in $pH$ corresponds to a $10$-fold change in $[H^{+}]$.
Example: If $[H^{+}] = 1 \times 10^{-2} \ M$,then $pH = 2$. If $[H^{+}] = 1 \times 10^{-3} \ M$,then $pH = 3$.