The mathematical expression of PH is

Analysis

The question asks for the correct mathematical definition of pH. In chemistry, pH is a measure of the acidity or basicity of an aqueous solution, specifically representing the concentration of hydrogen ions (or hydronium ions, $H_3O^+$) in that solution.

Concept Explanation

The pH scale is defined as the negative logarithm (base 10) of the hydrogen ion concentration. Mathematically, this is expressed as:

or, since $H^+$ ions exist in water as hydronium ions ($H_3O^+$):

Using the laws of logarithms, specifically the rule that $-\log(x) = \log(\frac{1}{x})$, we can rewrite the expression:

Step-by-Step Solution

1. Identify the standard definition: $pH = -\log_{10}[H_3O^+]$.
2. Apply logarithm rules: The negative sign in front of the log allows us to take the reciprocal of the value inside the bracket.
3. Transform the expression: $-\log_{10}[H_3O^+]$ becomes $\log_{10}\left(\frac{1}{[H_3O^+]}\right)$.
4. Compare with options: • Option A: $\log_{10}[OH^-]$ (This relates to pOH, not pH). • Option B: $\log_{10}\frac{1}{[H_3O^+]}$ (This matches our derived expression). • Option C: $\log_{10}[H_3O^+]$ (This is missing the negative sign or the reciprocal). • Option D: $\log_{10}\frac{1}{[OH^-]}$ (This relates to pOH).

Exam Tip

Always remember that "p" in pH stands for "power" or "potential," which mathematically translates to the negative log ($-\log$). If you see a "p" in front of a concentration (like pOH or pKa), it always means $-\log$ of that concentration. If you prefer to write it without the negative sign, you must use the reciprocal ($1/concentration$).