The concentration of hydronium ions in a cup of black coffee is `1.3 xx 10^(-5)` M. What will be the pH of the coffee?

Options

A. 4.89
B. 3.89
C. 2.2
D. None of these

Answer

To find the pH of a solution when the concentration of hydronium ions \([H_3O^+]\) is given, we can use the formula for pH:

\[ \text{pH} = -\log_{10} [H_3O^+] \]

### Step-by-Step Solution:

1. **Identify the concentration of hydronium ions**:
The concentration of hydronium ions in the coffee is given as \(1.3 \times 10^{-5}\) M.

2. **Substitute the value into the pH formula**:
We will substitute the concentration into the pH formula:
\[ \text{pH} = -\log_{10} (1.3 \times 10^{-5}) \]

3. **Calculate the logarithm**:
To calculate \(-\log_{10} (1.3 \times 10^{-5})\), we can break it down:
\[ \text{pH} = -\log_{10} (1.3) - \log_{10} (10^{-5}) \]
We know that \(-\log_{10} (10^{-5}) = 5\).

4. **Find \(-\log_{10} (1.3)\)**:
Using a calculator or logarithm table, we find:
\[ -\log_{10} (1.3) \approx -0.1139 \]

5. **Combine the results**:
Now, substituting back into the pH equation:
\[ \text{pH} = -0.1139 + 5 = 4.8861 \]

6. **Round the pH value**:
Typically, pH is rounded to two decimal places:
\[ \text{pH} \approx 4.89 \]

### Final Answer:
The pH of the coffee is approximately **4.89**.