Question

A voltaic cell utilizes the following reaction: $$ 2 \mathrm{Fe}^{3+}(a q)+\mathrm{H}_{2}(g) \rightarrow \rightarrow 2 \mathrm{Fe}^{2+}(a q)+2 \mathrm{H}^{+}(a q) $$ (a) What is the emf of this cell under standard conditions? (b) What is the emf for this cell when \(\left[\mathrm{Fe}^{3+}\right]=2.50 \mathrm{M}\), \(P_{\mathrm{H}_{2}}=0.85 \mathrm{~atm},\left[\mathrm{Fe}^{2+}\right]=0.0010 M\), and the \(\mathrm{pH}\) in both compartments is \(5.00 ?\)

Short answer

The emf under standard conditions for the given voltaic cell is 0.77 V. When the concentrations and pressure are modified as mentioned in the problem statement, the emf for the cell under non-standard conditions is approximately 0.83 V.

Step-by-step solution

Find the balanced half-reactions

Write the balanced half-cell reactions for the overall redox reaction given above: \(Fe^{3+}(aq) + e^- \rightarrow Fe^{2+}(aq)\) \(2H^+(aq) + 2e^- \rightarrow H_2(g)\)

Find the standard electrode potentials

Looking up the standard reduction potentials for the half-reactions in a reference table, we find: \( E^0_{Fe^{3+}/Fe^{2+}} = 0.77 V \) \( E^0_{H^+/H_2} = 0.00 V \)

Calculate the emf under standard conditions

To calculate the emf for the cell, subtract the standard reduction potential of the anode from the standard reduction potential of the cathode using the Nernst Equation: \(E^0_{cell} = E^0_{cathode} - E^0_{anode}\) \(E^0_{cell} = E^0_{Fe^{3+}/Fe^{2+}} - E^0_{H^+/H_{2}}\) \(E^0_{cell} = 0.77 V - 0.00 V\) \(E^0_{cell} = 0.77 V\) So the emf under standard conditions is 0.77 V.

Find the new concentrations and pressure

Now we have to calculate the emf for the non-standard conditions, using the concentrations and pressure given in the problem: \([Fe^{3+}]=2.50 M\) \([Fe^{2+}]=0.0010 M\) \(P_{H_2}=0.85 atm\) \(pH=5.00\) (from which we can find \([H^+]\) concentration)

Calculate the H+ concentration

Given the pH value, we can calculate the concentration of \(H^+\) ions as: \([H^+] = 10^{-pH}\) \([H^+] = 10^{-5.00}\) \([H^+] = 1.00 \times 10^{-5} M\)

Apply the Nernst Equation

Using the Nernst Equation, we can calculate the emf (E) for the cell under non-standard conditions: \( E = E^0_{cell} - \frac{0.0592}{n} \log{Q} \) Here, n is the number of moles of electrons transferred (which is 2 in this case), and Q is the reaction quotient.

Calculate the reaction quotient (Q)

The reaction quotient (Q) can be calculated as follows: \(Q = \frac{[Fe^{2+}]^2[H^+]^2}{[Fe^{3+}]^2}\) Substitute the given concentrations: \( Q = \frac{(0.0010)^2(1.00 \times 10^{-5})^2}{(2.50)^2} \)

Calculate the emf under non-standard conditions

Plugging the values of \(E^0_{cell}\), n, and Q into the Nernst Equation, we can calculate the emf for the non-standard conditions: \( E = 0.77 V - \frac{0.0592}{2} \log{Q} \) \( E = 0.77 V - \frac{0.0592}{2} \log{\frac{(0.0010)^2(1.00 \times 10^{-5})^2}{(2.50)^2}} \) Upon solving: \( E \approx 0.83 V\) The emf of this cell under the non-standard conditions is approximately 0.83 V.

Key concepts

Nernst Equation

Understanding the Nernst Equation is crucial for predicting how electrical potential, or electromotive force (emf), changes with concentration in electrochemical cells. It incorporates thermodynamics into electrochemistry, effectively linking the chemical reactivity of the cell's components with the voltage it can produce.

The general form of the Nernst Equation is: \[ E = E^0 - \frac{0.0592}{n} \log{Q} \]Here, \(E\) represents the cell potential under non-standard conditions, \(E^0\) is the standard cell potential, \(n\) is the number of moles of electrons transferred in the redox reaction, and \(Q\) is the reaction quotient. This equation adjusts the standard cell potential to account for the actual concentrations of reactants and products in the cell.

For students working with this equation, it is essential to remember that the log term accounts for the discrepancy between standard conditions (usually 1 M concentration and 1 atm pressure for gases) and the actual conditions. Also, the factor 0.0592 is derived from the constants of the equation at 298 K (25°C). If the temperature differs, this factor needs adjustment according to the formula \(\frac{0.0592}{T/298}\).

Standard Reduction Potential

The standard reduction potential, often symbolized as \(E^0\), is a measure of the tendency of a chemical species to gain electrons and thereby be reduced. Every half-reaction in an electrochemical cell has a standard reduction potential, which can be found in tables compiled through experimental measurements. These values are always provided at standard conditions, which are 25°C, 1 atm pressure for gases, and 1 M solutions.

When calculating the overall cell potential under standard conditions, you combine the standard reduction potentials of the two half-reactions. The cell potential is the difference between the cathode's and the anode's potentials:

\[E^0_{cell} = E^0_{cathode} - E^0_{anode}\]

For the given exercise, the standard reduction potential for \(Fe^{3+}/Fe^{2+}\) was given as 0.77 V, and for \(H^+/H_2\) as 0.00 V, since hydrogen's standard reduction potential is designated as the zero points for measuring other potentials. During the emf calculation under standard conditions, the value for the anode is subtracted from that of the cathode, providing a measure of the maximum potential difference between the two electrodes.

Reaction Quotient (Q)

The reaction quotient, \(Q\), represents the ratio of the concentrations of the reaction products to the reactants, each raised to the power of their stoichiometric coefficients. In the context of the cell reaction in the exercise, the reaction quotient is pivotal for understanding the direction in which the reaction is proceeding and how far from equilibrium it might be.

The formula for the reaction quotient looks like this:

\[Q = \frac{[\text{Products}]}{[\text{Reactants}]}\]

In the given voltaic cell case, the products are \(Fe^{2+}\) ions and \(H^+\) ions, while the reactants are \(Fe^{3+}\) ions. The reaction quotient fits into the Nernst Equation to account for non-standard conditions, which is crucial for accurate emf calculations when actual concentrations are different from the 1 M standard. Being familiar with calculating \(Q\) is fundamental for students, as it allows them to adjust the Nernst Equation to determine the cell potential at any given point in the reaction.

Electrochemistry

Electrochemistry is the branch of chemistry that deals with the relationships between electricity and chemical reactions. It encompasses the study of both generating electrical energy from chemical reactions and using electricity to induce chemical changes. Voltaic (or galvanic) cells, like the one presented in the exercise, generate electricity through spontaneous redox reactions. These cells are the basis for batteries and are of immense importance in everyday life, from powering handheld devices to vehicles.

Electrochemical processes are governed by a set of fundamental principles, which include the concepts we've discussed: the Nernst Equation, standard reduction potentials, and reaction quotients. Understanding these principles is essential for predicting the behavior of electrochemical systems and for applying this knowledge in practical applications such as energy storage, corrosion prevention, and electroplating metals.