Question

The triatomic molecular ion \(\mathrm{H}_{3}^{+}\) was first detected and characterized by J. J. Thomson using mass spectrometry. Use the bond energy of \(\mathrm{H}_{2}(432 \mathrm{~kJ} / \mathrm{mol})\) and the proton affinity of \(\mathrm{H}_{2}\left(\mathrm{H}_{2}+\mathrm{H}^{+} \longrightarrow \mathrm{H}_{3}^{+} ; \Delta H=-337 \mathrm{~kJ} / \mathrm{mol}\right)\) to calculate the en thalpy of reaction for \(\mathrm{H}+\mathrm{H}+\mathrm{H}^{+} \longrightarrow \mathrm{H}_{3}^{+}\)

Short answer

The enthalpy of the reaction is 95 kJ/mol.

Step-by-step solution

- Understand the Information Given

We are given the bond energy of \(\text{H}_2 \) as 432 kJ/mol, and the enthalpy change for the reaction \( \text{H}_2 + \text{H}^+ \rightarrow \text{H}_3^+ \) as -337 kJ/mol. These values will be used to calculate the enthalpy of the target reaction \( \text{H} + \text{H} + \text{H}^+ \rightarrow \text{H}_3^+ \).

- Determine the Energy Required to Break One Molecule of \(\text{H}_2\)

To break one molecule of \(\text{H}_2\), 432 kJ/mol of energy is required. This represents the energy needed to split \(\text{H}_2\) into two hydrogen atoms.

- Calculate the Energy Change for Producing \(\text{H} + \text{H}\)

Divide the bond energy of \(\text{H}_2\) by 2 since breaking one \( \text{H}_2\) molecule gives two hydrogen atoms: \(\frac{432 \text{ kJ}}{2} = 216 \text{ kJ}\) per hydrogen atom.

- Combine Two \(\text{H}\) Atoms and One Proton

Now form \(\text{H}_2\) from two H atoms, which releases -432 kJ/mol. Then add \(\text{H}^+\) (proton) to \(\text{H}_2\) to form \(\text{H}_3^+\) with an enthalpy change of -337 kJ/mol.

- Calculate Total Enthalpy Change

Combine the energy values: \( \text{Energy required to break one \(\text{H}_2\)} + \text{Enthalpy change to form \(\text{H}_3^+\)} = 432 \text{ kJ} - 337 \text{ kJ} = 95 \text{ kJ} \).

- Final Verification

Add the energies: \( 216 \text{ kJ} (to break \(\text{H}_2\)) + (-337 \text{ kJ} (to add \(\text{H}^+\) to \(\text{H}_2\)) = 95 \text{ kJ} \).

Key concepts

Bond Energy

Bond energy is the measure of the amount of energy required to break a chemical bond between atoms in a molecule. For example, the bond energy of \(H_2\) is 432 kJ/mol. This means that it takes 432 kJ to break one mole of hydrogen molecules (\text{H}_2\text\text) into individual hydrogen atoms (H).

In the exercise, we need to consider this bond energy to understand how much energy is required to break \(H_2\) into two H atoms. This is crucial for further calculations, as it sets the stage to determine the total energy change involved in forming \(H_3^+\) from \(H + H + H^+\).

Proton Affinity

Proton affinity refers to the energy change when a proton (\text{H}^+\text) is added to a molecule. In simpler terms, it’s a measure of how strongly a molecule attracts a proton. For \(H_2\), the proton affinity is represented by the enthalpy change of the reaction: \(H_2 + H^+ \rightarrow H_3^+\).

The problem states that the proton affinity of \(H_2\) is \(-337 kJ/mol\), meaning when a proton is added to \(H_2\), energy is released. This negative value indicates an exothermic reaction, where energy is given off.

In our exercise, this information helps us understand the energy change when \(H_2\) combines with \(H^+\) to form \(H_3^+\). This value is crucial for calculating the overall enthalpy change.

Mass Spectrometry

Mass spectrometry is an analytical technique used to measure the mass-to-charge ratio of ions. It's a powerful tool in detecting and characterizing molecules and ions. The instrument produces ions from the sample and sorts them based on their mass-to-charge ratios to produce signals for analysis.

J. J. Thomson used mass spectrometry to first detect \(H_3^+\), the triatomic molecular ion. Detection through mass spectrometry confirmed the existence of \(H_3^+\), providing insights into its formation and stability. This technique supports our understanding of the formation of \(H_3^+\) and helps in calculating the enthalpy change by analyzing the ions involved.

Enthalpy Change

Enthalpy change (∆H) is the difference in heat content between reactants and products in a chemical reaction at constant pressure. It is a crucial concept in understanding reaction energetics.

If ∆H is negative, the reaction is exothermic (releases heat), whereas if it’s positive, the reaction is endothermic (absorbs heat).

To calculate the enthalpy change for the formation of \H_3^+\ from \(H + H + H^+\), we consider the bond energy of \(H_2\) and the proton affinity of \(H_2\), as outlined in the steps.

For instance, we first determine the energy needed to break \(H_2\) (432 kJ/mol), and then the energy change when adding \(H^+\) to \(H_2\) (-337 kJ/mol). Combining these values gives the total enthalpy change, helping us understand the energy dynamics of the reaction.