Question

For the given cell :
\(\mathrm{Cu(s)\,|\,Cu^{2+}(C_1\,M)\,||\,Cu^{2+}(C_2\,M)\,|\,Cu(s)}\)
change in Gibbs energy (\(\Delta G\)) is negative, if :

• A. \(\mathrm{C_2=\sqrt{2}\,C_1}\)
• B. \(\mathrm{C_2=\frac{C_1}{\sqrt{2}}}\)
• C. \(\mathrm{C_1=2C_2}\)
• D. \(\mathrm{C_1=C_2}\)

Hint:

This is a concentration cell with \(E^\circ=0\); \(\Delta G \lt 0\) needs \(E_{cell} \gt 0\), i.e. the cathode (right) concentration must be higher.

Answer 1

The Correct Option is A

Left electrode (anode): \(\mathrm{Cu\rightarrow Cu^{2+}(C_1)+2e^-}\). Right electrode (cathode): \(\mathrm{Cu^{2+}(C_2)+2e^-\rightarrow Cu}\). Overall: \(\mathrm{Cu^{2+}(C_2)\rightarrow Cu^{2+}(C_1)}\), \(E^\circ=0\).
Nernst: \(E=-\frac{0.059}{2}\log\frac{C_1}{C_2}=\frac{0.059}{2}\log\frac{C_2}{C_1}\).
\(\Delta G=-nFE \lt 0\) requires \(E \gt 0\), i.e. \(C_2 \gt C_1\).
(1) \(C_2=\sqrt2C_1 \gt C_1\): yes. (2) \(C_2=C_1/\sqrt2 \lt C_1\): no. (3) \(C_1=2C_2\Rightarrow C_2 \lt C_1\): no. (4) \(C_1=C_2\): \(E=0,\ \Delta G=0\): no.
Answer: (1) \(C_2=\sqrt{2}\,C_1\)