Question

(a) Given that the enthalpy change associated with the addition of \(\mathrm{H}^{+}(\mathrm{g})\) to \(\mathrm{H}_{2} \mathrm{O}(\mathrm{g})\) is \(-690 \mathrm{kJ} \mathrm{mol}^{-1},\) and \(\Delta_{\mathrm{hyd}} H^{\circ}\left(\mathrm{H}^{+}, \mathrm{g}\right)=-1091 \mathrm{kJ} \mathrm{mol}^{-1},\) calculate the enthalpy change associated with the solvation of \(\left[\mathrm{H}_{3} \mathrm{O}\right]^{+}(\mathrm{g})\) in water. (b) Outline how the nickel-metal hydride battery works, giving equations for the reactions at each electrode during charging and discharging.

Short answer

(a) The solvation enthalpy of [H₃O]⁺(g) in water is -401 kJ/mol. (b) The NiMH battery discharges by oxidizing MH to M and reducing NiO(OH) to Ni(OH)₂; it charges via the reverse reactions.

Step-by-step solution

Understanding the Given Enthalpy Changes

We are provided with two different enthalpy changes: 1. The enthalpy change for the addition of gaseous H⁺ to gaseous H₂O to form gaseous hydronium ion [H₃O]⁺, which is given as -690 kJ/mol. 2. The standard enthalpy of hydration for gaseous H⁺, which is -1091 kJ/mol.

Calculate Solvation Enthalpy of [H₃O]⁺(g) in Water

To find the solvation enthalpy of [H₃O]⁺(g) in water, use the formula:\[\Delta_{ ext{solvation}} H^{ heta} = \Delta_{ ext{hyd}} H^{ heta} - (\text{Enthalpy of reaction})\]Plug in the values:\[\Delta_{ ext{solvation}} H^{ heta} = -1091 \, \text{kJ/mol} - (-690 \, \text{kJ/mol})\]\[\Delta_{ ext{solvation}} H^{ heta} = -401 \, \text{kJ/mol}\]

Nickel-Metal Hydride Battery Discharge Reaction

During discharge, the battery reactions at the electrodes are:- At the anode (negative electrode), the oxidation reaction: \[ \text{MH} + \text{OH}^- \rightarrow \text{M} + \text{H}_2\text{O} + \text{e}^- \]- At the cathode (positive electrode), the reduction reaction: \[ \text{NiO}(OH) + \text{H}_2\text{O} + \text{e}^- \rightarrow \text{Ni}(OH)_2 + \text{OH}^- \]

Nickel-Metal Hydride Battery Charge Reaction

During charging, the reactions are reversed:- At the anode (negative electrode), the reduction reaction: \[ \text{M} + \text{H}_2\text{O} + \text{e}^- \rightarrow \text{MH} + \text{OH}^- \]- At the cathode (positive electrode), the oxidation reaction: \[ \text{Ni}(OH)_2 + \text{OH}^- \rightarrow \text{NiO}(OH) + \text{H}_2\text{O} + \text{e}^- \]

Key concepts

Solvation Enthalpy

Solvation enthalpy refers to the energy change observed when a solute dissolves in a solvent. In our context, we are particularly interested in the solvation enthalpy of the hydronium ion \([H_3O]^+\) in water. Think of it as the energy released or absorbed when gaseous hydronium ions are surrounded by water molecules to form a solution. To compute this, we consider two main energy changes:

• The enthalpy change for forming \([H_3O]^+\) from \( ext{H}^+\) and \( ext{H}_2 ext{O}\) in the gas phase.
• The hydration enthalpy of \( ext{H}^+\) ions, which is the energy change when \( ext{H}^+\) ions in the gas phase dissolve in water.

By using the given enthalpy values, we find the solvation enthalpy using the formula: \[\Delta_{\text{solvation}} H^{\theta} = -1091 \, \text{kJ/mol} - (-690 \, \text{kJ/mol}) \\Delta_{\text{solvation}} H^{\theta} = -401 \, \text{kJ/mol}\]This negative value indicates that the process is exothermic, meaning energy is released when solvation occurs.

Nickel-Metal Hydride Battery

Nickel-metal hydride batteries, often abbreviated as NiMH batteries, are rechargeable power sources. They are prevalent in everyday devices like digital cameras and hybrid vehicles. These batteries operate based on redox (reduction-oxidation) reactions, which involve the transfer of electrons.During discharge:- At the anode, there is an oxidation reaction involving metal hydride (MH), resulting in the release of an electron \(\text{e}^-\).\[\text{MH} + \text{OH}^- \rightarrow \text{M} + \text{H}_2\text{O} + \text{e}^- \\]- At the cathode, nickel oxy-hydroxide \(\text{NiO}(\text{OH})\) gets reduced, consuming the electron.\[\text{NiO}(\text{OH}) + \text{H}_2\text{O} + \text{e}^- \rightarrow \text{Ni}(\text{OH})_2 + \text{OH}^- \\]During charging:- The reactions reverse: the anode sees a reduction reaction while the cathode undergoes oxidation.

• At the anode: \(\text{M} + \text{H}_2\text{O} + \text{e}^- \rightarrow \text{MH} + \text{OH}^-\)
• At the cathode: \(\text{Ni}(\text{OH})_2 + \text{OH}^- \rightarrow \text{NiO}(\text{OH}) + \text{H}_2\text{O} + \text{e}^-\)

This alternating cycle of reactions is what charges the battery and allows it to power electronic devices.

Reaction Equations

Reaction equations are fundamental in chemistry as they express chemical changes using symbols and numbers. They quantitatively describe the transformation of reactants to products.For example, the general form of a chemical reaction can be represented as:\[\text{A} + \text{B} \rightarrow \text{C} + \text{D} \\]In these equations, reactants are substances that undergo chemical changes, leading to the formation of products. Let's see how this applies in the context of NiMH batteries:

• During discharge, the reactants at the anode are \(\text{MH}\) and \(\text{OH}^-\), which transform into \(\text{M}\), \(\text{H}_2\text{O}\), and \(\text{e}^-\).
• At the cathode, \(\text{NiO}(\text{OH})\) and \(\text{H}_2\text{O}\) react with \(\text{e}^-\) to form \(\text{Ni}(\text{OH})_2\) and \(\text{OH}^-\).
• The reverse occurs during charging.

These equations help in understanding what materials are consumed and what products are formed during chemical reactions, which is crucial in predicting the behavior of chemical systems.