Question

Suppose that the gas-phase reaction \(2 \mathrm{NO}(g)+\mathrm{O}_{2}(g)\) \(\longrightarrow 2 \mathrm{NO}_{2}(g)\) were carried out in a constant-volume container at constant temperature. Would the measured heat change represent \(\Delta H\) or \(\Delta E ?\) If there is a difference, which quantity is larger for this reaction? Explain.

Short answer

In this gas-phase reaction at constant temperature and volume, the measured heat change represents both the change in internal energy (ΔE) and the change in enthalpy (ΔH). There is no difference between ΔH and ΔE for this reaction because both quantities are equal to the measured heat change.

Step-by-step solution

Recall the relationships between heat, work, internal energy, and enthalpy

To solve this problem, we need to recall the relationships between heat (q), work (w), internal energy (∆E), and enthalpy (∆H). Here are the relevant relationships: 1. ∆E = q + w 2. ∆H = ∆E + P∆V Let's use these relationships to determine whether the measured heat change represents ∆H or ∆E and which quantity is larger for this reaction.

Determine whether the measured heat change represents ∆H or ∆E

In this reaction, the temperature and volume are held constant. When the volume is constant, there is no pressure-volume work done (w = 0). Therefore, according to the first relationship (∆E = q + w), we have ∆E = q, since w = 0. So in this case, the measured heat change represents the change in internal energy (∆E).

Determine if there is a difference between ∆H and ∆E

To determine if there is a difference between ∆H and ∆E, let's use the second relationship (∆H = ∆E + P∆V). As we determined in Step 2, ∆E = q. So, we can write: ∆H = q + P∆V Since the volume is constant, ∆V = 0. Therefore, the term P∆V = 0. Now, the relationship becomes: ∆H = q This means that there is no difference between ∆H and ∆E in this reaction because both values are equal to the measured heat change.

Conclusion

In conclusion, the measured heat change under constant temperature and volume conditions for the gas-phase reaction represents both the change in internal energy (ΔE) and the change in enthalpy (ΔH). There is no difference between ∆H and ∆E for this reaction because both quantities are equal to the measured heat change.

Key concepts

Enthalpy

In the realm of thermodynamics, **enthalpy** is a vital concept that helps us understand how energy changes in chemical processes. Enthalpy, denoted as \(\Delta H\), includes both the internal energy of a system and the energy associated with the volume and pressure of the system. In essence, it's a measure of the total heat content of a system.

When a reaction occurs at constant pressure, the change in enthalpy (\(\Delta H\)) matches the heat exchanged with the surroundings. This is the heat we typically think of when dealing with everyday chemical reactions. However, for reactions in a constant-volume container, like the one described in our exercise, enthalpy's role changes slightly.

**Key Points about Enthalpy:**

• Often associated with reactions at constant pressure.
• Represents the heat absorbed or released.
• At constant volume, \(\Delta H = \Delta E\) if there is no work due to volume change.

Internal Energy

**Internal Energy**, denoted as \(\Delta E\), represents all the energy within a system. In chemical reactions, it's the total of kinetic and potential energies of the molecules involved. When we talk about internal energy in this context, we're considering:

- The vibration of atoms and molecules.
- The energy stored in chemical bonds.

For a gas-phase reaction in a constant volume container, like the one given, the crucial point about internal energy is how it changes only through the heat added or removed, not through work. Since the volume does not change, the work done by the system is zero. This means that any heat change directly translates to a change in internal energy.

**Salient Details on Internal Energy**:

• Internal energy changes (\(\Delta E = q\) when volume is constant).
• No work is done when a reaction occurs at constant volume.
• Reflects energy stored in chemical bonds and motion of particles.

Gas-phase Reaction

In the world of chemistry, **gas-phase reactions** involve reactants and products that are all in the gaseous state. These reactions can be quite dynamic due to the high energy and movement of gas molecules. The particular gas-phase reaction in our exercise involves nitrogen monoxide \((\text{NO})\) reacting with oxygen \((\text{O}_2)\) to form nitrogen dioxide \((\text{NO}_2)\). This type of reaction is common in various atmospheric processes and combustion events.

During such reactions, the role of pressure and volume becomes noticeable, especially in context with energy changes. Because gases are compressible, how volume and pressure are controlled impacts the reaction. In our scenario, as it's done at constant volume, the only energy changes come from heat, influencing internal energy and enthalpy as explained earlier.

**Insights on Gas-phase Reactions:**

• All participants in the reaction are gases.
• Highly dependent on temperature and volume conditions.
• In constant volume reactions, energy changes are solely due to heat.