To find the standard free-energy change ($\Delta G^\circ$) for the reaction, we can use the following relationship between the standard free-energy change and the standard cell potential ($E^\circ_{\text{cell}}$):

$\Delta G^\circ = -nFE^\circ_{\text{cell}}$

where:

• $n$ is the number of moles of electrons transferred in the balanced reaction.
• $F$ is the Faraday constant ($F = 96485 \, \text{C/mol}$).
• $E^\circ_{\text{cell}}$ is the standard cell potential.

First, we need to determine the number of electrons transferred ($n$) in the reaction:

$\text{Cr}_2\text{O}_7^{2-}(aq) + 3\text{Sn}^{2+}(aq) + 14\text{H}^+(aq) \rightarrow 2\text{Cr}^{3+}(aq) + 3\text{Sn}^{4+}(aq) + 7\text{H}_2\text{O}(l)$

The change in oxidation states are:

• Chromium changes from +6 in $\text{Cr}_2\text{O}_7^{2-}$ to +3 in $\text{Cr}^{3+}$. Each Cr atom gains 3 electrons, so for 2 Cr atoms, a total of 6 electrons are gained.
• Tin changes from +2 in $\text{Sn}^{2+}$ to +4 in $\text{Sn}^{4+}$. Each Sn atom loses 2 electrons, so for 3 Sn atoms, a total of 6 electrons are lost.

Thus, the total number of electrons transferred ($n$) is 6.

Now we can use the formula to calculate $\Delta G^\circ$:

$\Delta G^\circ = -nFE^\circ_{\text{cell}}$ $\Delta G^\circ = -6 \times 96485 \, \text{C/mol} \times 1.21 \, \text{V}$

Let's calculate this:

$\Delta G^\circ = -6 \times 96485 \times 1.21$ $\Delta G^\circ = -699603.1 \, \text{J/mol}$

Since the answer needs to be in volts (V), we need to convert joules to volts. This is a bit unconventional since $\Delta G^\circ$ is typically expressed in energy units (Joules or kJ), but in the context of electrical work, 1 J = 1 V·C.

However, if we interpret the question as needing the value in standard energy units (Joules or kJ), we can express it as follows:

$\Delta G^\circ = -699.603 \, \text{kJ/mol}$

Thus, the standard free-energy change for this reaction at 25°C is $-699.603 \, \text{kJ/mol}$.

Do you want more details or have any questions about this calculation?

Here are some related questions you might find interesting:

1. How do you determine the number of moles of electrons transferred in a redox reaction?
2. What is the Faraday constant, and how is it used in electrochemistry?
3. How do you balance redox reactions in acidic and basic solutions?
4. What is the significance of the standard cell potential in electrochemical cells?
5. How does temperature affect the standard free-energy change in a reaction?

Tip: When working with electrochemical reactions, always make sure to balance the redox reactions properly and identify the number of electrons transferred to accurately calculate the free-energy changes.