Sure, let's go through the detailed, step-by-step solution to determine the correct formula for calculating the \( pH \) depending on the \( H^+ \) concentration \(\left(\left[ H^+\right]\right)\).
1. Understanding pH: The pH of a solution is a measure of its acidity or basicity. It is defined as:
[tex]\[
pH = -\log \left(\left[ H^+\right]\right)
\][/tex]
where \(\left[ H^+\right]\) is the concentration of hydrogen ions (or hydronium ions) in the solution.
2. Given Information: You measured a \( pH \) of 7 in a solution that has \( 10^{-7} \) moles per liter of dissociated \( H_2O \) (H2O is ionized to \( H^+ \) and \( OH^-\)). This implies that:
[tex]\[
-\log \left(10^{-7}\right) = 7
\][/tex]
3. Analyzing the Options:
a) \( pH = \log \left(\left[ H^+\right] \times 10^{-7}\right) \)
- This suggests taking a logarithm of the product of the hydrogen ion concentration and \( 10^{-7} \). This doesn't fit with our definition of pH as it should involve just \(\left[ H^+\right]\).
b) \( pH = -\log \left(1 \times 10^7\right) \)
- This option is incorrect as it ignores the concentration of \( H^+ \) ions; also, \( 10^7 \) is the inverse of the concentration for a neutral solution, hence, resulting in incorrect calculation.
c) \( pH = \log \left(1 \times 10^7\right) \)
- This option similarly ignores the concentration of \( H^+ \) ions and provides a formulation which is not aligned with how pH is actually defined and calculated.
d) \( pH = -\log \left(\left[ H^+\right]\right) \)
- This option correctly matches the universally accepted definition of pH where it is the negative logarithm of the hydrogen ion concentration.
4. Correct Answer: Given what we've examined, the correct formula is:
[tex]\[
pH = -\log \left(\left[ H^+\right]\right)
\][/tex]
Thus, the correct option is:
d) \( pH = -\log \left(\left[ H^+\right]\right) \).
This defines the pH in a manner consistent with the commonly accepted scientific definition.
