Question

What will be the emf for the given cell?
\[Pt\mid {H_2}{\text{ }}\left( {g,{P_1}} \right)\mid {H^ + }\left( {aq} \right)\mid \mid {H_2}\left( {g,{P_2}} \right)\mid Pt\]
A. $\dfrac{{RT}}{F}\ln \dfrac{{{P_1}}}{{{P_2}}} \\$
B. $\dfrac{{RT}}{{2F}}\ln \dfrac{{{P_1}}}{{{P_2}}} \\$
C. $\dfrac{{RT}}{F}\ln \dfrac{{{P_2}}}{{{P_1}}} \\$
D. $none{\text{ }}of{\text{ }}these $

Answer 1

Hint:The emf i.e. electromotive force of a cell can be defined as the maximum potential difference that exists between the two electrodes of a cell. In other words, it can be referred to as the net voltage that exists between the reduction and oxidation half-reactions. The emf of a cell is majorly utilised to identify if the given electrochemical cell is galvanic or not.

Complete answer:
Even under the non-standard conditions, Nernst equation can be used to determine cell potentials of the electrochemical cells. The Nernst equation is generally employed in order to calculate the cell potentials of an electrochemical cell at a given temperature, reactant concentration and pressure. The Nernst equation is written below:
$E = {E^o} - \dfrac{{RT}}{{zF}}\ln Q$
Where, E is reduction potential, Eo is standard potential, R is universal gas constant, T is temperature (in Kelvin), z is ion charge (i.e. moles of electrons), F is Faraday constant and Q is reaction quotient
In the present case, the redox chemical equations can be written as follows:
$
  2{H^ + } + 2{e^ - } \to {H_2}({P_2}) \\
  {H_2}({P_1}) \to 2{H^ + } + 2{e^ - }
 $
(From here, we get to know that z = 2)
Thus, the overall reaction can be written as:
${H_2}({P_1}) \to {H_2}({P_2})$
As a result, Nernst equation can be written as:
$
 E = {E^o} - \dfrac{{RT}}{{zF}}\ln \dfrac{{{P_2}}}{{{P_1}}} = 0 - \dfrac{{RT}}{{zF}}\ln \dfrac{{{P_2}}}{{{P_1}}} = - \dfrac{{RT}}{{zF}}\ln \dfrac{{{P_2}}}{{{P_1}}} \\
 = \dfrac{{RT}}{{zF}}\ln \dfrac{{{P_1}}}{{{P_2}}} \\
 $(Since, ${E^o}_{{H^ + }/{H_2}} = 0$)
Since z = 2, the final Nernst equation is $E = \dfrac{{RT}}{{2F}}\ln \dfrac{{{P_1}}}{{{P_2}}}$

Hence, the correct answer is Option B.

Note:
Nernst equation basically depicts the direct relationship between the electrode potential of a half cell and the temperature. Thus, it means that as the temperature of the half cell increases, electrode potential also increases.