At certain temperature, the `H^(+)` ion concentration of water is `4 xx 10^(-7)` 'M then the value of `K_(w)`, at the same temperature is

Options

A. `10^(-14) M^(2)`
B. `4xx10^(-14)M^(2)`
C. `1.6xx10^(-13) M^(2)`
D. `4xx10^(-7)M^(2)`

Answer

To solve the problem of finding the value of \( K_w \) (the ionic product of water) at a certain temperature given the concentration of \( H^+ \) ions, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the relationship of \( K_w \)**:
The ionic product of water (\( K_w \)) is defined as:
\[ K_w = [H^+][OH^-] \]
For pure water, the concentration of \( H^+ \) ions is equal to the concentration of \( OH^- \) ions.

2. **Given concentration of \( H^+ \)**:
The problem states that the concentration of \( H^+ \) ions is:
\[ [H^+] = 4 \times 10^{-7} \, M \]

3. **Determine the concentration of \( OH^- \)**:
Since \( [H^+] = [OH^-] \) in pure water, we also have:
\[ [OH^-] = 4 \times 10^{-7} \, M \]

4. **Calculate \( K_w \)**:
Substitute the values of \( [H^+] \) and \( [OH^-] \) into the equation for \( K_w \):
\[ K_w = [H^+][OH^-] = (4 \times 10^{-7})(4 \times 10^{-7}) \]

5. **Perform the multiplication**:
\[ K_w = 16 \times 10^{-14} \]

6. **Express in scientific notation**:
Convert \( 16 \times 10^{-14} \) to proper scientific notation:
\[ K_w = 1.6 \times 10^{-13} \, M^2 \]

### Final Answer:
Thus, the value of \( K_w \) at the given temperature is:
\[ K_w = 1.6 \times 10^{-13} \, M^2 \]