Question

Which of the following is true for a spontaneous galvanic cell?

• A. \(E^\circ_{cell}0,\ Q_c>K_c\)
• B. \(E^\circ_{cell}=0,\ \Delta G^\circ=0,\ Q_c=K_c\)
• C. \(E^\circ_{cell}<0,\ \Delta G^\circ<0,\ Q_c<K_c\)
• D. \(E^\circ_{cell}>0,\ \Delta G^\circ<0,\ Q_c<K_c\)

Hint:

For electrochemical cells, remember the shortcut: \[ E^\circ_{cell}>0 \Rightarrow \text{Spontaneous reaction} \] \[ E^\circ_{cell}=0 \Rightarrow \text{Equilibrium} \] \[ E^\circ_{cell}<0 \Rightarrow \text{Non-spontaneous reaction} \]

Answer 1

The Correct Option is D

Concept: A galvanic cell is an electrochemical cell in which a redox reaction occurs spontaneously to produce electrical energy. The spontaneity of the reaction depends upon the sign of Gibbs free energy change and the value of cell potential. The important relation connecting Gibbs free energy and cell potential is: \[ \Delta G^\circ = -nFE^\circ_{cell} \] where, \[ n = \text{number of electrons transferred} \] \[ F = \text{Faraday constant} \] \[ E^\circ_{cell} = \text{standard cell potential} \] For a spontaneous process: \[ \Delta G^\circ < 0 \] Hence, \[ E^\circ_{cell} > 0 \] Thus, a spontaneous galvanic cell must always have a positive standard cell potential.

Step 1: Understanding the sign of \(E^\circ_{cell}\). From the equation \[ \Delta G^\circ = -nFE^\circ_{cell} \] we observe that: \[ \Delta G^\circ < 0 \] only when \[ E^\circ_{cell} > 0 \] because \(n\) and \(F\) are always positive constants. Therefore, for spontaneous operation of the galvanic cell: \[ E^\circ_{cell}>0 \]

Step 2: Understanding the relation between \(Q_c\) and \(K_c\). The reaction quotient \(Q_c\) indicates the current state of the reaction, whereas the equilibrium constant \(K_c\) indicates the equilibrium state. For a reaction to proceed spontaneously in the forward direction: \[ Q_c < K_c \] This means the system has not yet reached equilibrium and hence the reaction proceeds forward naturally. At equilibrium: \[ Q_c = K_c \] and at equilibrium: \[ E_{cell}=0 \] Therefore, a spontaneous galvanic cell must satisfy: \[ Q_c<K_c \]

Step 3: Checking all the given options.
• Option (1): \(E^\circ_{cell}<0\) and \(\Delta G^\circ>0\) correspond to a non-spontaneous reaction. Hence incorrect.
• Option (2): \(E^\circ_{cell}=0\) and \(\Delta G^\circ=0\) represent equilibrium condition, not a spontaneous galvanic cell. Hence incorrect.
• Option (3): \(\Delta G^\circ<0\) is correct for spontaneity, but \(E^\circ_{cell}<0\) contradicts the relation \(\Delta G^\circ=-nFE^\circ_{cell}\). Hence incorrect.
• Option (4): \(E^\circ_{cell}>0,\ \Delta G^\circ<0,\ Q_c<K_c\) satisfies all conditions for spontaneity. Hence correct. Final Answer: \[ E^\circ_{cell}>0,\qquad \Delta G^\circ<0,\qquad Q_c<K_c \] Therefore, option \((4)\) is the correct answer.