For the Mg-Ag cell, how many times the difference between the EMF of the cell and its standard EMF will change if concentration of `Mg^(2+)` ions is changed from 0.1 M to 0.01 M and that of `Ag^(+)` ions is chagned from 0.5 M to 0.25 M?

Explanation

To solve the problem regarding the Mg-Ag cell and the changes in EMF due to the concentration changes of ions, we can follow these steps:

### Step 1: Write the Nernst Equation
The Nernst equation for the cell is given by:
\[ E = E^\circ - \frac{0.0591}{n} \log \left( \frac{[\text{Mg}^{2+}]}{[\text{Ag}^{+}]^2} \right) \]
where \(E\) is the cell potential, \(E^\circ\) is the standard cell potential, \(n\) is the number of moles of electrons transferred, and the concentrations are those of the ions involved in the reaction.

### Step 2: Identify the Initial and Final Concentrations
- Initial concentration of \(\text{Mg}^{2+}\) is \(0.1 \, M\).
- Final concentration of \(\text{Mg}^{2+}\) is \(0.01 \, M\).
- Initial concentration of \(\text{Ag}^{+}\) is \(0.5 \, M\).
- Final concentration of \(\text{Ag}^{+}\) is \(0.25 \, M\).

### Step 3: Calculate the Change in EMF for Both Cases
1. **For the first case (initial concentrations)**:
\[ E_1 = E^\circ - \frac{0.0591}{2} \log \left( \frac{0.1}{(0.5)^2} \right) \]

2. **For the second case (final concentrations)**:
\[ E_2 = E^\circ - \frac{0.0591}{2} \log \left( \frac{0.01}{(0.25)^2} \right) \]

### Step 4: Calculate the Difference in EMF
The difference between the standard EMF and the cell EMF for both cases can be expressed as:
\[ E^\circ - E_1 \quad \text{and} \quad E^\circ - E_2 \]

### Step 5: Find the Ratios
To find how many times the difference changes, we need to evaluate:
\[ \frac{E^\circ - E_2}{E^\circ - E_1} \]

### Step 6: Substitute Values
Substituting the values into the logarithmic terms:
- For \(E_1\):
\[ E_1 = E^\circ - \frac{0.0591}{2} \log \left( \frac{0.1}{0.25} \right) = E^\circ - \frac{0.0591}{2} \log(0.4) \]

- For \(E_2\):
\[ E_2 = E^\circ - \frac{0.0591}{2} \log \left( \frac{0.01}{0.0625} \right) = E^\circ - \frac{0.0591}{2} \log(0.16) \]

### Step 7: Calculate the Logarithmic Values
Using logarithmic properties:
\[ \log(0.4) \approx -0.3979 \quad \text{and} \quad \log(0.16) \approx -0.7959 \]

### Step 8: Calculate the Change in EMF
Now, we can calculate the differences:
\[ E^\circ - E_1 \quad \text{and} \quad E^\circ - E_2 \]

### Step 9: Determine the Ratio
Finally, we find the ratio of the differences:
\[ \frac{E^\circ - E_2}{E^\circ - E_1} \approx 2 \]

### Conclusion
The difference between the EMF of the cell and its standard EMF will change by a factor of 2 when the concentrations of \(\text{Mg}^{2+}\) and \(\text{Ag}^{+}\) are altered as specified.

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