Calculate the emf of the cell in which the following reaction takes place: $Ni(s) + 2Ag^+(0.002 M) ightarrow Ni^{2+}(0.160 M) + 2Ag(s)$
Given that $E^0{cell} = 1.05 V$. Also, calculate the standard Gibbs energy change ($\Deltar G^0$) and equilibrium constant ($Kc$) for the reaction at 298 K. (Use $F = 96500 C mol^{-1}$, $\log(0.08) = -1.097$, $R = 8.314 J K^{-1} mol^{-1}$)

Question

Calculate the emf of the cell in which the following reaction takes place: $Ni(s) + 2Ag^+(0.002 M) ightarrow Ni^{2+}(0.160 M) + 2Ag(s)$
Given that $E^0{cell} = 1.05 V$. Also, calculate the standard Gibbs energy change ($\Deltar G^0$) and equilibrium constant ($Kc$) for the reaction at 298 K. (Use $F = 96500 C mol^{-1}$, $\log(0.08) = -1.097$, $R = 8.314 J K^{-1} mol^{-1}$)

Accepted Answer

Step-by-Step Working: Identification of parameters from the given cell reaction: The given cell reaction is: $Ni(s) + 2Ag^+(aq) ightarrow Ni^{2+}(aq) + 2Ag(s)$ Number of electrons transferred ($n$) = 2 Standard cell potential ($E^0{cell}$) = $1.05 V$ Concentration of $Ni^{2+}$ ions $Ni^{2+} = 0.160 M$ Concentration of $Ag^+$ ions $Ag^+ = 0.002 M$ Temperature ($T$) = $298 K$ Faraday constant ($F$) = $96500 C mol^{-1}$ Calculation of Cell emf ($E{cell}$) using Nernst Equation:
The Nernst equation for the cell is: $$E{cell} = E^0{cell} - \frac{0.0591}{n} \log \frac{Ni^{2+}}{Ag^+^2}$$ 
Substitute the given values into the equation: $$E{cell} = 1.05 - \frac{0.0591}{2} \log \frac{0.160}{(0.002)^2}$| $$E{cell} = 1.05 - 0.02955 \log \frac{0.160}{0.000004}$$ $$E{cell} = 1.05 - 0.02955 \log (40000)$$ $$\log(40000) = \log(4 	imes 10^4) = \log 4 + 4 \log 10 = 0.6021 + 4(1) = 4.6021$$ 
Now, substitute the logarithmic value: $$E{cell} = 1.05 - 0.02955 	imes 4.6021$$ $$E{cell} = 1.05 - 0.1360 = 0.914 V$$ (Alternative approximation method gives around $0.91 V$) Calculation of Standard Gibbs Energy Change ($\Deltar G^0$):
The relation between standard Gibbs energy and standard cell potential is: $$\Deltar G^0 = -n F E^0{cell}$$
Substitute the values: $$\Deltar G^0 = -2 	imes 96500 C mol^{-1} 	imes 1.05 V$$ $$\Deltar G^0 = -203000 	imes 1.05 J mol^{-1} = -213150 J mol^{-1}$$ $$\Deltar G^0 = -213.15 kJ mol^{-1}$$ Calculation of Equilibrium Constant ($Kc$):
The relation between standard cell potential and equilibrium constant at 298 K is: $$\log Kc = \frac{n E^0{cell}}{0.0591}$$
Substitute the values: $$\log Kc = \frac{2 	imes 1.05}{0.0591} = \frac{2.10}{0.0591} = 35.533$$
Taking antilog: $$Kc = 	ext{antilog}(35.533) = 3.41 	imes 10^{35}$$

Step-by-Step Working:

1. Identification of parameters from the given cell reaction:

2. Calculation of Cell emf ($E_{cell}$) using Nernst Equation:\nThe Nernst equation for the cell is:

3. Calculation of Standard Gibbs Energy Change ($\Delta_r G^0$):\nThe relation between standard Gibbs energy and standard cell potential is:

4. Calculation of Equilibrium Constant ($K_c$):\nThe relation between standard cell potential and equilibrium constant at 298 K is: