Question

Given the listed standard electrode potentials, what is \(E^{\circ}\) for the cell : \(4 \mathrm{BiO}^{+}(a q)+3 \mathrm{~N}_{2} \mathrm{H}_{5}^{+}(a q) \longrightarrow 4 \mathrm{Bi}(s)+3 \mathrm{~N}_{2}(g)+4 \mathrm{H}_{2} \mathrm{O}(l)+7 \mathrm{H}^{+}(a q)\) \(\mathrm{N}_{2}(g)+5 \mathrm{H}^{+}(a q)+4 e^{-} \longrightarrow \mathrm{N}_{2} \mathrm{H}_{5}^{+}(a q), E^{\circ}=-0.23 \mathrm{~V}\) \(\mathrm{BiO}^{+}(a q)+2 \mathrm{H}^{+}(a q)+3 e^{-} \longrightarrow \mathrm{Bi}(s)+\mathrm{H}_{2} \mathrm{O}(l), \quad E^{\circ}=+0.32 \mathrm{~V}\) (a) \(+0.55\) (b) \(+0.34\) (c) \(+1.88\) (d) \(+0.09\)

Short answer

+0.55

Step-by-step solution

Write the half-reactions

Identify and write the two half-reactions that occur in the cell. The oxidation half-reaction is: \(N_2(g) + 5H^+(aq) + 4e^- \longrightarrow N_2H_5^+(aq), \quad E^\circ = -0.23\, V\) The reduction half-reaction is: \(BiO^+(aq) + 2H^+(aq) + 3e^- \longrightarrow Bi(s) + H_2O(l), \quad E^\circ = +0.32\, V\)

Balance the electrons in the half-reactions

To combine the half-reactions into a full reaction, they must transfer the same number of electrons. Multiply the oxidation half-reaction by 3 to match the 12 electrons transferred in the reduction half-reaction. The oxidation half-reaction becomes: \(3N_2(g) + 15H^+(aq) + 12e^- \longrightarrow 3N_2H_5^+(aq), \quad E^\circ = -0.23\, V\)

Combine the half-reactions

Add the two half-reactions together, ensuring that the electrons cancel out. The combined balanced reaction will be: \(4BiO^+(aq) + 3N_2(g) + 11H^+(aq) + 12e^- \longrightarrow 4Bi(s) + 3N_2H_5^+(aq) + 4H_2O(l) + 7H^+(aq)\)

Calculate the cell potential

Calculate the standard cell potential \(E^\circ\) for the full reaction using the standard electrode potentials of the half-reactions. \(E^\circ_{cell} = E^\circ_{cathode} - E^\circ_{anode}\) \(E^\circ_{cell} = 0.32\, V - (-0.23\, V) = 0.55\, V\)

Key concepts

Electrochemistry

Electrochemistry is a branch of chemistry that studies the interaction between electrical energy and chemical change. It includes the study of both spontaneous and non-spontaneous chemical reactions that involve the transfer of electrons. These reactions occur in devices called electrochemical cells, which can be categorized into two types: galvanic cells, which generate electrical energy from spontaneous chemical reactions, and electrolytic cells, which use electrical energy to drive non-spontaneous reactions.

An important application of electrochemistry is in the development of batteries and fuel cells, which store and provide electrical energy through chemical reactions. Other applications include metal plating, electroplating, and corrosion prevention. Understanding the principles of electrochemistry is key to advancements in these areas, as well as in broader fields such as materials science and energy technology.

Cell Potential Calculation

Cell potential, often represented as \( E^{\circ}_{cell} \), measures the driving force behind the electrochemical reaction occurring in an electrochemical cell. It is a key concept in determining the direction of electron flow and the ability of a cell to do electrical work. In a typical calculation for standard cell potential, one employs the standard electrode potentials for the cathode (reduction half-reaction) and the anode (oxidation half-reaction).

The cell potential can be calculated using the formula: \[ E^{\circ}_{cell} = E^{\circ}_{cathode} - E^{\circ}_{anode} \]. The standard electrode potentials used in this calculation are measured under standard conditions: 1 M concentration, 1 atm pressure, and 25°C. A positive cell potential indicates a spontaneous reaction, whereas a negative cell potential signifies a non-spontaneous reaction. These calculations are crucial in predicting the feasibility and direction of the reactions in galvanic cells.

Half-Reaction Method

The half-reaction method is an approach used to balance redox reactions, which involve the transfer of electrons between species. Redox reactions are split into two half-reactions: one that involves the loss of electrons (oxidation) and one that involves the gain of electrons (reduction).

During the balancing process, each half-reaction is balanced separately for mass and charge. Then, they are combined in a way that ensures the number of electrons lost in the oxidation half-reaction is equal to the number gained in the reduction half-reaction. This ensures that charge is conserved when the reaction takes place. The ability to effectively use the half-reaction method is essential in electrochemistry, as it allows for the elucidation and understanding of intricate redox processes occurring within an electrochemical cell.

Balancing Redox Reactions

Balancing redox reactions is a fundamental skill in electrochemistry, essential for understanding how reactions proceed and for calculations involving cell potentials. Redox reactions are balanced to ensure that the number of atoms of each element and the total charge is the same on both sides of the reaction equation.

To achieve this balance, one often follows a systematic process: separate the reaction into half-reactions; balance the atoms of each element other than oxygen and hydrogen; balance oxygen atoms by adding \( H_2O \) molecules; balance hydrogen atoms by adding \( H^+ \) ions; balance the charges by adding electrons; and finally, multiply the half-reactions by appropriate coefficients so that the electrons cancel when the half-reactions are added together. Mastery of balancing redox reactions ensures accurate reflection of the matter and charge conservation in chemical processes.