Electrochemical Cell Potential Analysis

The question asks how to obtain the standard electrode potential, E°(M+/M), from a plot involving the cell potential (E_cell) of the cell SHE||MX|M, where MX is an ideal dilute solution with molality m, and temperature T.

The cell notation SHE||MX|M represents an electrochemical cell where the Standard Hydrogen Electrode (SHE) acts as one electrode, and the M|MX electrode acts as the other. The double vertical line (||) indicates a salt bridge or porous membrane separating the two half-cells. The reaction at the SHE (anode) is typically:

\begin{equation*} \frac{1}{2}\text{H}_2\text{(g, 1 atm)} \rightarrow \text{H}^+\text{(aq, } a_{\text{H}^+} \text{)} + \text{e}^- \quad E^\circ = 0 \text{ V} \end{equation*}

Assuming a standard SHE where $a_{\text{H}^+} = 1$ and $P_{\text{H}_2} = 1$ atm, its potential is 0 V.

The reaction at the M|MX electrode (cathode) is:

\begin{equation*} \text{M}^+\text{(aq, } a_{\text{M}^+} \text{)} + \text{e}^- \rightarrow \text{M(s)} \end{equation*}

The potential of the M|MX electrode is given by the Nernst equation:

\begin{equation*} E_{\text{M}^+/\text{M}} = E^\circ_{\text{M}^+/\text{M}} - \frac{RT}{nF} \ln\left(\frac{a_{\text{M}}}{a_{\text{M}^+}}\right) \end{equation*}

For the reaction M$^+$ + e$^-$ → M(s), the number of electrons transferred, n, is 1. The activity of the solid metal, $a_{\text{M}}$, is taken as 1. Thus, the equation becomes:

\begin{equation*} E_{\text{M}^+/\text{M}} = E^\circ_{\text{M}^+/\text{M}} - \frac{RT}{1F} \ln\left(\frac{1}{a_{\text{M}^+}}\right) = E^\circ_{\text{M}^+/\text{M}} + \frac{RT}{F} \ln(a_{\text{M}^+}) \end{equation*}

For an ideal dilute solution of MX, the activity of the cation M$^+$ can be approximated by its molality, m. Therefore, $a_{\text{M}^+} \approx m$. Using the relationship $\ln(x) = 2.303 \log_{10}(x)$, we get:

\begin{equation*} E_{\text{M}^+/\text{M}} \approx E^\circ_{\text{M}^+/\text{M}} + \frac{2.303 RT}{F} \log_{10}(m) \end{equation*}

The cell potential, E_cell, is the difference between the cathode potential and the anode potential:

\begin{equation*} E_{\text{cell}} = E_{\text{cathode}} - E_{\text{anode}} = E_{\text{M}^+/\text{M}} - E_{\text{SHE}} \end{equation*}

Assuming the SHE is standard (potential 0 V):

\begin{equation*} E_{\text{cell}} \approx \left(E^\circ_{\text{M}^+/\text{M}} + \frac{2.303 RT}{F} \log_{10}(m)\right) - 0 \end{equation*}

So, the relationship between E_cell and molality m at temperature T is approximately:

\begin{equation*} E_{\text{cell}} \approx E^\circ_{\text{M}^+/\text{M}} + \left(\frac{2.303 R}{F}\right) T \log_{10}(m) \end{equation*}

This equation shows that E_cell is linearly related to the term $(T \log_{10} m)$, assuming $E^\circ_{\text{M}^+/\text{M}}$ and the constant $\frac{2.303 R}{F}$ are constant.

If we plot E_cell on the y-axis against $(T \log_{10} m)$ on the x-axis, the equation is in the form $y = mx + c$, where:

• $y = E_{\text{cell}}$
• $x = T \log_{10}(m)$
• $m = \frac{2.303 R}{F}$ (slope)
• $c = E^\circ_{\text{M}^+/\text{M}}$ (y-intercept)

Therefore, the standard electrode potential E°(M+/M) can be obtained as the intercept of the plot of E_cell against $(T \log m)$.

Let's check the given options based on this derivation:

• Option 1: E_cell against (T log m). This matches our finding.
• Option 2: E_cell against (1/T log m). This does not fit the derived linear relationship.
• Option 3: E_cell against (T√m). This does not fit the derived linear relationship which involves log(m).
• Option 4: E_cell against ($\sqrt m / T$). This does not fit the derived linear relationship.

The plot of E_cell versus (T log m) should yield a straight line, and the value of E°(M+/M) is the intercept on the E_cell axis (when T log m = 0, which happens when m=1 at any T, or when T=0 at any m, practically evaluated by extrapolating to log m = 0, i.e., m=1). However, often such plots are extrapolated to m=0 (infinite dilution) to account for deviations from ideal behavior, but the ideal dilute solution assumption simplifies it to a plot against T log m.

Thus, plotting E_cell against (T log m) allows for the determination of E°(M+/M).