Statement A: “The electrical work that a reaction can perform at constant pressure and temperature is equal to the reaction Gibbs energy.”

For a galvanic cell operating reversibly at constant temperature and pressure, $\Delta G = -nFE_{\text{cell}}$ and the maximum electrical work is $w_{\text{elec,max}} = -\Delta G$ So the magnitude of electrical work equals the decrease in Gibbs energy, but it is not equal to $\Delta G$ directly because of sign convention.

Hence, Statement A is incorrect.

Statement B: “$E_{\text{cell}}^{\circ}$ is dependent on the pressure.”

Standard cell potential is related to standard Gibbs energy: $\Delta G^{\circ} = -nF E_{\text{cell}}^{\circ}$ Since standard Gibbs energy depends on temperature and on the defined standard states, and for gases the standard state involves pressure, $E_{\text{cell}}^{\circ}$ can depend on pressure.

Hence, Statement B is correct.

Statement C: $\frac{dE_{\text{cell}}^{\theta}}{dT} = \frac{\Delta_r S^{\theta}}{nF}$

We use $\Delta G^{\circ} = -nF E_{\text{cell}}^{\circ}$ Differentiate w.r.t. $T$: $\frac{d(\Delta G^{\circ})}{dT} = -nF\frac{dE_{\text{cell}}^{\circ}}{dT}$ But, $\left(\frac{dG}{dT}\right)_P = -S \Rightarrow \frac{d(\Delta G^{\circ})}{dT} = -\Delta_r S^{\circ}$ Therefore, $-\Delta_r S^{\circ} = -nF\frac{dE_{\text{cell}}^{\circ}}{dT}$ $\frac{dE_{\text{cell}}^{\circ}}{dT} = \frac{\Delta_r S^{\circ}}{nF}$

Hence, Statement C is correct.

Statement D: “A cell is operating reversibly if the cell potential is exactly balanced by an opposing source of potential difference.”

This is the condition of reversible operation: when the opposing potential is equal to the cell emf, current tends to zero and the process is reversible.

Hence, Statement D is correct.

Counting incorrect statements

• A: Incorrect
• B: Correct
• C: Correct
• D: Correct

Therefore, the number of incorrect statements = 1.