Calculate the concentration of an unknown solution in a concentration cell. A concentration cell consisting of two hydrogen electrodes (P H2atm), where the cathode is a standard hydrogen electrode and the anode solution has an unknown pH, has a cell voltage of 0.181 V. What is the pH in the unknown solution? Assume the temperature of the solutions is 298 K
The Correct Answer and Explanation is :
To calculate the pH of the unknown solution in a concentration cell, we can use the Nernst equation, which relates the cell potential to the concentrations (or activities) of the ions involved in the electrochemical reaction. In this case, the cell consists of two hydrogen electrodes, one of which is a standard hydrogen electrode (SHE) and the other is an unknown pH solution.
Step 1: Write the Nernst equation
The Nernst equation for a concentration cell involving hydrogen ions (H⁺) is:
[
E = E^\circ – \frac{0.0592}{n} \log \frac{[H^+]{\text{anode}}}{[H^+]{\text{cathode}}}
]
Where:
- (E) is the observed cell potential (in volts),
- (E^\circ) is the standard electrode potential for the hydrogen electrode (which is 0 V),
- (n) is the number of electrons transferred (which is 2 for the hydrogen ion reduction half-reaction),
- ([H^+]{\text{anode}}) and ([H^+]{\text{cathode}}) are the hydrogen ion concentrations at the anode and cathode, respectively.
Step 2: Apply the Nernst equation
For the SHE (standard hydrogen electrode), the pH is 0 (because ([H^+] = 1 \, \text{M})). The concentration of hydrogen ions at the cathode is 1 M, and at the anode, the concentration is related to the pH of the unknown solution, as ([H^+] = 10^{-\text{pH}}).
The equation becomes:
[
E = 0 – \frac{0.0592}{2} \log \frac{10^{-\text{pH}}}{1}
]
Simplifying:
[
E = -0.0296 \log (10^{-\text{pH}})
]
Since (\log (10^{-\text{pH}}) = -\text{pH}), the equation becomes:
[
E = 0.0296 \times \text{pH}
]
Step 3: Solve for pH
We know that the cell voltage (E) is 0.181 V. Plugging this value into the equation:
[
0.181 = 0.0296 \times \text{pH}
]
Solving for pH:
[
\text{pH} = \frac{0.181}{0.0296} = 6.11
]
Conclusion:
The pH of the unknown solution is 6.11.
Explanation:
This calculation uses the Nernst equation, which describes the relationship between the electrode potential of a concentration cell and the concentration (or activity) of ions in the solutions. In this case, the potential difference of 0.181 V corresponds to a difference in the hydrogen ion concentrations between the anode (unknown pH) and the cathode (pH 0). By using the Nernst equation, we calculate the pH of the unknown solution as 6.11, indicating it is slightly acidic.