At any temperature , the proton concentration of water is

Options

A. `10^(-14) M`
B. `Kw`
C. `gt 10^(-7) M`
D. `sqrt(K_(w))`

Explanation

To find the proton concentration of water at any temperature, we can follow these steps:

### Step 1: Understand the Ionic Product of Water
The ionic product of water (Kw) is defined as the product of the concentrations of hydrogen ions (H⁺) and hydroxide ions (OH⁻) in water. This can be expressed as:
\[ K_w = [H^+][OH^-] \]

### Step 2: Recognize the Equality of Ion Concentrations in Pure Water
In pure water, the concentration of hydrogen ions is equal to the concentration of hydroxide ions:
\[ [H^+] = [OH^-] \]

### Step 3: Substitute into the Ionic Product Equation
Since the concentrations of H⁺ and OH⁻ are equal, we can substitute [H⁺] for [OH⁻] in the expression for Kw:
\[ K_w = [H^+][H^+] = [H^+]^2 \]

### Step 4: Solve for the Concentration of H⁺
To find the concentration of H⁺ ions, we take the square root of both sides:
\[ [H^+] = \sqrt{K_w} \]

### Step 5: Consider the Value of Kw at Different Temperatures
The value of Kw changes with temperature. For example:
- At 25°C, \( K_w = 10^{-14} \)
- At 90°C, \( K_w = 10^{-12} \)

### Step 6: Calculate [H⁺] for Specific Temperatures
Using the values of Kw:
- At 25°C:
\[ [H^+] = \sqrt{10^{-14}} = 10^{-7} \, \text{mol/L} \]
- At 90°C:
\[ [H^+] = \sqrt{10^{-12}} = 10^{-6} \, \text{mol/L} \]

### Conclusion
At any temperature, the proton concentration in water can be expressed as:
\[ [H^+] = \sqrt{K_w} \]

### Final Answer
Thus, the proton concentration of water at any temperature is given by:
\[ [H^+] = \sqrt{K_w} \]