To find the magnitude of the cell potential when the concentration of \( M^+ \) is changed from 0.05 M to 0.0025 M, we can use the Nernst equation for a concentration cell. Here’s the step-by-step solution:
### Step 1: Understand the Nernst Equation
The Nernst equation is given by:
\[
E_{cell} = E^0 - \frac{RT}{nF} \ln Q
\]
Where:
- \( E_{cell} \) is the cell potential,
- \( E^0 \) is the standard cell potential,
- \( R \) is the universal gas constant (8.314 J/(mol·K)),
- \( T \) is the temperature in Kelvin,
- \( n \) is the number of moles of electrons transferred,
- \( F \) is Faraday's constant (96485 C/mol),
- \( Q \) is the reaction quotient.
### Step 2: Identify the Initial and Final Concentrations
We have:
- Initial concentration of \( M^+ \) (C1) = 0.05 M
- Final concentration of \( M^+ \) (C2) = 0.0025 M
### Step 3: Calculate the Reaction Quotient (Q)
For a concentration cell, the reaction quotient \( Q \) can be expressed as:
\[
Q = \frac{[M^+]_{anode}}{[M^+]_{cathode}}
\]
In this case, we can assume:
- At the anode: \( [M^+] = 0.0025 \, M \)
- At the cathode: \( [M^+] = 0.05 \, M \)
Thus,
\[
Q = \frac{0.0025}{0.05} = 0.05
\]
### Step 4: Calculate the Cell Potential Change
Using the Nernst equation, we can express the change in cell potential as:
\[
E_{cell} = E^0 - \frac{0.059}{n} \log Q
\]
Since we are comparing two states, we can set up the ratio:
\[
\frac{E_{cell2}}{E_{cell1}} = \frac{\log [M^+]_{2}}{\log [M^+]_{1}}
\]
Substituting the values:
\[
\frac{E_{cell2}}{E_{cell1}} = \frac{\log(0.0025)}{\log(0.05)}
\]
### Step 5: Calculate the Logarithmic Values
Calculating the logarithmic values:
\[
\log(0.0025) \approx -2.6
\]
\[
\log(0.05) \approx -1.3
\]
### Step 6: Calculate the Ratio of Cell Potentials
Now, substituting these values into the ratio:
\[
\frac{E_{cell2}}{70 \, \text{mV}} = \frac{-2.6}{-1.3} = 2
\]
Thus,
\[
E_{cell2} = 2 \times 70 \, \text{mV} = 140 \, \text{mV}
\]
### Final Answer
The magnitude of the cell potential when the concentration of \( M^+ \) is changed to 0.0025 M is **140 mV**.
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