For the reaction \(\operatorname{Co}(\mathrm{s})+\mathrm{Ni}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Co}^{2+}(\mathrm{aq})+\) \(\mathrm{Ni}(\mathrm{s}), E_{\mathrm{cell}}^{\circ}=0.03 \mathrm{V} .\) If cobalt metal is added to an aqueous solution in which \(\left[\mathrm{Ni}^{2+}\right]=1.0 \mathrm{M},\) (a) the reaction will not proceed in the forward direction at all; (b) the displacement of \(\mathrm{Ni}(\mathrm{s})\) from the \(\mathrm{Ni}^{2+}(\mathrm{aq})\) will go to completion; (c) the displacement of \(\mathrm{Ni}(\mathrm{s})\) from the solution will proceed to a considerable extent, but the reaction will not go to completion; (d) there is no way to predict how far the reaction will proceed.

Step by step solution

01

Recall the Nernst equation

The Nernst equation provides a relation between the cell potential, standard cell potential and the concentrations of the reactants and products. It is given as \(E_{\mathrm{cell}}=E_{\mathrm{cell}}^{\circ}-\frac{0.0591}{n} \log Q\), where \(Q\) is the reaction quotient and \(n\) is the number of electrons transferred in the reaction.

02

Find the reaction quotient (Q)

Here, the reaction quotient \(Q\) for the given reaction \(\operatorname{Co}(\mathrm{s})+ \mathrm{Ni}^{2+}(\mathrm{aq}) \longrightarrow \mathrm{Co}^{2+}(\mathrm{aq})+ \mathrm{Ni}(\mathrm{s})\) would be expressed as \(Q = \frac{[\text{Co}^{2+}]}{[\text{Ni}^{2+}]}.\) At the beginning of the reaction, the concentration of \(\mathrm{Co}^{2+}\) is 0, so Q=0.

03

Apply the Nernst Equation

Substitute \(E_{\mathrm{cell}}^{\circ}\), \(n\) and \(Q\) into the Nernst equation. Here, \(E_{\mathrm{cell}}^{\circ} = 0.03 \, V\), \(n = 2\), and \(Q = 0\). This will give: \[E_{\mathrm{cell}} = 0.03 \, V - \frac{0.0591}{2} \log (0).\] This leads to a mathematical problem because the logarithm of 0 is undefined.

04

Resolve the mathematical issue

When Q = 0, \(E_{cell}\) is at its maximum. The Nernst equation demonstrates that as the reaction progresses (as Q increases), the cell potential will decrease. Therefore, as long as \(E_{cell}\) is greater than 0, the reaction is spontaneous.

05

Determine the direction of the reaction

Since the initial \(E_{cell}\) was greater than zero (0.03V), the reaction is spontaneous and will proceed. However, it will not proceed to completion. This is because the Nernst equation shows that as the reaction proceeds and the concentration of \(\mathrm{Ni}^{2+}\) decreases (with a corresponding increase in \(\mathrm{Co}^{2+}\)), the cell potential \(E_{cell}\) will decrease, eventually to a value less than 0. At this point, the reaction will reach equilibrium and the net reaction will stop.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!