Effect of temperature on voltage of a galvanic cell can't use Nernst equation? [closed]

Question

Is it worth investigating the effect of temperature on EMF in a galvanic cell? I suspect the change will be very very small and maybe impossible to test precisely in a high school lab.
This article said the variation of E with temperature can be find with E(T2)=E(T1)+dE/dT(T2-T1) where dE/dT=R/nF (lnQ) but how is this different to using the Nernst equation?

Answer 1

TL;DR; version:

Testing the EMF dependence on temperature does make sense, because it is not the matter just of the term $\dfrac{RT}{nF} \ln {Q}$. But you cannot use the Nernst equation directly as it does not explicitly enumerate the temperature dependence of standard reduction potentials.

The derivative of $\text{EMF}$ by absolute temperature depends on:

• the current component thermodynamic activities(concentrations,pressures, fugacities) ($\dfrac{RT}{nF} \ln {Q}$)
• the reaction entropy ( $\Delta_\text{r} S^{\circ}$)
• the difference of the heat capacity of products and reactants ($\Delta C_p$)

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Detailed evaluation:

\begin{align} \text{EMF}(T) &= \Delta E^{\circ}(T) - \frac{RT}{nF} \ln{Q} =\\ &= - \frac{\Delta_\text{r} G^{\circ}(T)}{nF} - \frac{RT}{nF} \ln{Q} =\\ &= - \frac{\Delta_\text{r} G^{\circ}(T) + RT \ln{Q} }{nF}\\ &= - \frac{\Delta_\text{r} H^{\circ}(T) - T \left( \Delta_\text{r} S^{\circ}(T) - R \ln{Q}\right) }{nF}\\ &\approx - \frac{\Delta_\text{r} H^{\circ}_{T_1} +\Delta C_p(T - T_1) - T \left( \Delta_\text{r} S^{\circ}_{T_1} + \Delta C_p \ln {\frac{T}{T_1}} - R \ln{Q}\right) }{nF} \tag{1} \end{align}

The key influence has the parameter $\Delta C_p$, what is the difference between the total heat capacity of reaction products and reactants. The approximation in the last step assumes constant molar heat capacities of involved components. If such a condition is not met, things would get complicated because of the need of integration of the functions $\int_{T_1}^{T_2}{\Delta C_p(T)\text{d}T}$ and $\int_{T_1}^{T_2}{\dfrac{\Delta C_p (T)}{T}\text{d}T}$

$$\frac{\text{d}(\text{EMF}(T))}{\text{d}T} \approx -\frac 1{nF} \left( \Delta C_p - \left( \Delta_\text{r} S^{\circ}_{T_1} + \Delta C_p \ln {\frac{T}{T_1}} - R \ln{Q}\right) - T \left( \frac{\Delta C_p }{ T} \right) \right)= \\ = \frac 1{nF} \left( \Delta_\text{r} S^{\circ}_{T_1} + \Delta C_p \ln {\frac{T}{T_1}} - R \ln{Q}\right) \tag{2} \label{2}$$

In the case where $\Delta C_p \approx 0$, the equation gets simplified to:

\begin{align} \text{EMF}(T) = - \frac{\Delta_\text{r} H^{\circ}_{T_1} - T \left( \Delta_\text{r} S^{\circ}_{T_1} - R \ln{Q}\right) }{nF} \tag{3} \end{align}

$$\frac{\text{d}(\text{EMF}(T))}{\text{d}T} = \frac 1{nF} \left( \Delta_\text{r} S^{\circ}_{T_1} - R \ln{Q}\right) \tag{4} $$

As the direct conclusion of the equation \eqref{2} comes the TL;DR; answer.