The hydrogen ion concentration of a `10^(-8) M HCl` aqueous soultion at `298 K(K_(w)=10^(-14))` is

Options

A. `9.525xx10^(-8) M`
B. `1.0xx10^(-8) M`
C. `1.0xx10^(-6) M`
D. `1.0525xx10^(-7) M`

Explanation

To find the hydrogen ion concentration of a `10^(-8) M HCl` aqueous solution at `298 K`, we follow these steps:

### Step 1: Understand the dissociation of HCl
Hydrochloric acid (HCl) is a strong acid that completely dissociates in water:
\[ \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \]
Since the concentration of HCl is `10^(-8) M`, the initial concentration of hydrogen ions from HCl is also `10^(-8) M`.

### Step 2: Consider the contribution of water
In addition to the HCl dissociation, water also undergoes autoionization:
\[ \text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^- \]
At 298 K, the ion product of water (\(K_w\)) is given as \(10^{-14}\):
\[ K_w = [\text{H}^+][\text{OH}^-] = 10^{-14} \]

### Step 3: Set up the equations
Let \(X\) be the concentration of hydrogen ions produced from the autoionization of water. Therefore, the total concentration of hydrogen ions in the solution will be:
\[ [\text{H}^+] = 10^{-8} + X \]
And since the concentration of hydroxide ions will also be \(X\), we can express the ion product of water as:
\[ K_w = (10^{-8} + X)(X) = 10^{-14} \]

### Step 4: Solve for X
Expanding the equation:
\[ 10^{-8}X + X^2 = 10^{-14} \]
This is a quadratic equation in \(X\):
\[ X^2 + 10^{-8}X - 10^{-14} = 0 \]

### Step 5: Use the quadratic formula
Using the quadratic formula \(X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\):
Here, \(a = 1\), \(b = 10^{-8}\), and \(c = -10^{-14}\):
\[ X = \frac{-10^{-8} \pm \sqrt{(10^{-8})^2 - 4(1)(-10^{-14})}}{2(1)} \]
\[ X = \frac{-10^{-8} \pm \sqrt{10^{-16} + 4 \times 10^{-14}}}{2} \]
\[ X = \frac{-10^{-8} \pm \sqrt{4.01 \times 10^{-14}}}{2} \]
\[ X = \frac{-10^{-8} \pm 2.0025 \times 10^{-7}}{2} \]

Calculating the positive root:
\[ X \approx \frac{1.9025 \times 10^{-7}}{2} \approx 0.95125 \times 10^{-7} \]

### Step 6: Calculate the total hydrogen ion concentration
Now substituting back to find the total concentration of hydrogen ions:
\[ [\text{H}^+] = 10^{-8} + X \]
\[ [\text{H}^+] = 10^{-8} + 0.95125 \times 10^{-7} \]
\[ [\text{H}^+] = 1.0525 \times 10^{-7} \, M \]

### Final Answer
Thus, the hydrogen ion concentration of the `10^(-8) M HCl` aqueous solution at `298 K` is:
\[ [\text{H}^+] \approx 1.0525 \times 10^{-7} \, M \]