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

Options

A. `1.0 xx 10^(-8) M`
B. `1.0 xx 10^(-6) M`
C. `1.0525 xx 10^(-7) M`
D. `9.525 xx 10^(-8) M`

Answer

To find the hydrogen ion concentration of a \(10^{-8} M\) HCl aqueous solution at \(298 K\) where \(K_w = 10^{-14}\), we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Dissociation of HCl:**
HCl is a strong acid and dissociates completely in water:
\[ \text{HCl} \rightarrow \text{H}^+ + \text{Cl}^- \]
Therefore, the initial concentration of hydrogen ions from HCl is \(10^{-8} M\).

2. **Considering the Contribution from Water:**
Water also dissociates slightly into hydrogen ions and hydroxide ions:
\[ \text{H}_2\text{O} \rightleftharpoons \text{H}^+ + \text{OH}^- \]
The concentration of hydroxide ions \([\text{OH}^-]\) in pure water at \(298 K\) is:
\[ [\text{H}^+][\text{OH}^-] = K_w = 10^{-14} \]
Since the concentration of hydrogen ions from HCl is \(10^{-8} M\), we can denote the concentration of hydrogen ions from water as \(x\).

3. **Setting Up the Equation:**
The total concentration of hydrogen ions \([\text{H}^+]\) in the solution will be:
\[ [\text{H}^+] = 10^{-8} + x \]
The concentration of hydroxide ions will also be \(x\). Therefore, we can write:
\[ K_w = (10^{-8} + x)(x) = 10^{-14} \]

4. **Expanding the Equation:**
Expanding the equation gives:
\[ 10^{-8}x + x^2 = 10^{-14} \]
Rearranging it results in:
\[ x^2 + 10^{-8}x - 10^{-14} = 0 \]

5. **Solving the Quadratic Equation:**
We can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) where \(a = 1\), \(b = 10^{-8}\), and \(c = -10^{-14}\):
\[ x = \frac{-10^{-8} \pm \sqrt{(10^{-8})^2 - 4 \cdot 1 \cdot (-10^{-14})}}{2 \cdot 1} \]
\[ x = \frac{-10^{-8} \pm \sqrt{10^{-16} + 4 \cdot 10^{-14}}}{2} \]
\[ x = \frac{-10^{-8} \pm \sqrt{4.01 \cdot 10^{-14}}}{2} \]
\[ x \approx \frac{-10^{-8} \pm 2 \cdot 10^{-7}}{2} \]
Taking the positive root:
\[ x \approx 0.95 \cdot 10^{-7} M \]

6. **Calculating Total Hydrogen Ion Concentration:**
Now we can find the total concentration of hydrogen ions:
\[ [\text{H}^+] = 10^{-8} + x = 10^{-8} + 0.95 \cdot 10^{-7} \]
\[ = 1.0525 \cdot 10^{-7} M \]

### Final Answer:
The hydrogen ion concentration of the \(10^{-8} M\) HCl aqueous solution at \(298 K\) is approximately:
\[ 1.0525 \times 10^{-7} M \]