Question

For the electrochemical cell, $Mg(s){\text{|M}}{{\text{g}}^{2 + }}(aq,1M){\text{||}}C{u^{2 + }}(aq,1M){\text{|}}Cu(s)$ the standard EMF of the cell is $2.70V$ at 300K. When the concentration of $M{g^{2 + }}$ is changed to $xM$, the cell potential changes to $2.67V$ at 300K. The value of $x$ is _______.

Answer 1

Hint: To answer this question, you need to recall the Nernst equation. The Nersnt equation relates the EMF, temperature and concentrations of chemical species undergoing redox reactions.
Formula used:
$E = {E^0} - \dfrac{{RT}}{{nF}}\ln \dfrac{{\left[ {M{g^{2 + }}} \right]}}{{\left[ {C{u^{2 + }}} \right]}}$
Where, $E$ is the EMF of the cell, ${E^0}$ is the standard cell potential, $n$ is the number of electrons transferred in the reaction, $F$ is Faraday constant, $R$ is the gas constant and $T$ is the temperature.

Complete step by step answer:
 The cell reaction is given by:$Mg(s) + C{u^{2 + }}(aq) \rightleftharpoons M{g^{2 + }}(aq) + Cu(s)$
Here, two electrons are transferred during the course of the reaction. So, $n = 2$.
We are given the concentration of magnesium ion as $x$.
Using the Nernst equation, $E = {E^0} - \dfrac{{RT}}{{nF}}\ln \dfrac{{\left[ {M{g^{2 + }}} \right]}}{{\left[ {C{u^{2 + }}} \right]}}$
$
   \Rightarrow 2.67 = 2.70 - \dfrac{{RT}}{{nF}}\ln \left( {\dfrac{x}{1}} \right) \\
   \Rightarrow \ln x = \dfrac{{0.03 \times nF}}{{RT}} \\
 $
Substituting the values, we get
$
   \Rightarrow \ln x = \dfrac{{0.03 \times 2 \times 11500}}{{300}} = 2.30 \\
   \Rightarrow \ln x = 2.30 \approx \ln 10 \\
  \therefore x = 10 \\
 $
The answer is $x = 10$.

Additional information:
Limitations of Nernst Equation:Nernst equation can be expressed directly in the terms of concentrations of constituents in dilute solutions. But at higher concentrations, the true activities of the ions become significant and therefore, must be used. This complicates the Nernst equation, as estimation of these non-ideal activities of ions requires complex experimental measurements. Also, the Nernst equation applies only when there is no net current flow through the electrode. The activity of ions at the electrode surface changes when current flows, and there are additional overpotential and resistive loss terms which contribute to the measured potential.

Note:
The Nersnt equation permits the extent of reaction between two redox systems to be calculated and is usually used to decide whether a particular reaction will go to completion or not. At equilibrium, the EMFs of the two half cells are equal. This enables equilibrium constant to be calculated and hence, the extent of the reaction.