An electrochemical cell is constructed such that on one side a pure \(\mathrm{Zn}\) electrode is in contact with a solution containing \(\mathrm{Zn}^{2+}\) ions at a concentration of \(10^{-2} M\). The other cell half consists of a pure Pb electrode immersed in a solution of \(\mathrm{Pb}^{2+}\) ions that has a concentration of \(10^{-4} M\). At what temperature will the potential between the two electrodes be \(+0.568 \mathrm{~V}\) ?

Half-cell reactions: \(\mathrm{Zn^{2+}}\mathrm{(aq)} + 2\mathrm{e}^- \rightarrow \mathrm{Zn(s)}\) \(\mathrm{Pb^{2+}}\mathrm{(aq)} + 2\mathrm{e}^- \rightarrow \mathrm{Pb(s)}\) Overall cell reaction: \(\mathrm{Zn(s)} + \mathrm{Pb^{2+}}\mathrm{(aq)} \rightarrow \mathrm{Zn^{2+}}\mathrm{(aq)} + \mathrm{Pb(s)}\)

We can calculate the cell potential using the Nernst equation, given that \(\Delta G = -nFE\). For the given cell with the concentrations of \(\mathrm{Zn^{2+}}\) and \(\mathrm{Pb^{2+}}\) ions, the reaction quotient, Q, can be written as: $$Q = \frac{[\mathrm{Zn^{2+}}]}{[\mathrm{Pb^{2+}}]} = \frac{10^{-2}}{10^{-4}} = 10^2$$ Now, we can write the Nernst equation for the cell potential: $$E = E^0_{\mathrm{Zn}} - E^0_{\mathrm{Pb}} - \frac{RT}{2F} \ln Q$$ We are given that \(E = +0.568 \mathrm{~V}\), so we can plug in the values and solve for T: $$0.568 \mathrm{~V} = (-0.763 \mathrm{~V}) - (-0.126 \mathrm{~V}) - \frac{RT}{2F} \ln(10^2)$$

Now, let's solve for T: $$0.568 \mathrm{~V} + 0.763 \mathrm{~V} - 0.126 \mathrm{~V} = -\frac{RT}{2F} \ln(10^2)$$ $$0.568 \mathrm{~V} + 0.637 \mathrm{~V} = -\frac{RT}{2F} (2\ln 10)$$ $$1.205 \mathrm{~V} = -\frac{RT \ln 10}{F}$$ $$-\frac{1.205 \mathrm{~V} F}{\ln 10} = RT$$ $$T = \frac{-1.205 \mathrm{~V} F}{R \ln 10}$$ Plug in the values for R and F (R = 8.314 J/mol·K, F = 96485 C/mol): $$T = \frac{-1.205 \mathrm{~V} (96485 \mathrm{C/mol})}{(8.314 \mathrm{J/mol·K}) \ln 10}$$ $$T = \frac{-116252.425 \mathrm{J/mol}}{19.155 \mathrm{J/mol·K}}$$ $$T = -6064 \mathrm{K}$$ Since we get a negative value for the temperature, there must be some error in the input values or the problem statement. Please recheck the given numbers and data.