Problem 123 - Chemistry An Atoms First Approach

Question

For the process \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g)\) at \(298 \mathrm{K}\) and 1.0 atm, \(\Delta H\) is more positive than \(\Delta E\) by 2.5 kJ/mol. What does the \(2.5 \mathrm{kJ} / \mathrm{mol}\) quantity represent?

Step-by-Step Solution

Answer

The 2.5 kJ/mol difference between ΔH and ΔE for the process \(\mathrm{H}_{2} \mathrm{O}(l) \longrightarrow \mathrm{H}_{2} \mathrm{O}(g)\) at 298 K and 1.0 atm represents the work done by the system on its surroundings as the water molecules transform from a liquid phase to gaseous phase, at constant pressure and temperature.

Step 1: Understanding the Terms

Enthalpy (H) and Internal Energy (E) are state functions that describe the energy content of a system. Here, we deal with their changes (ΔH and ΔE) in conversion from liquid water to water vapor. Enthalpy represents the amount of heat energy transferred at constant pressure and accounts for the work done by the system on its surroundings. On the other hand, internal energy represents the total energy within a system (including kinetic and potential energy). The relationship between ΔH, ΔE, and the work done during the process is: \( \Delta H = \Delta E + P \Delta V\) where P is the constant pressure and ΔV is the change in volume for the process.

Step 2: Analyze the Given Information

We are given that ΔH is 2.5 kJ/mol more positive than ΔE. We can rewrite the relationship in step 1 as: \( \Delta H - \Delta E = P \Delta V \) The 2.5 kJ/mol represents the difference between ΔH and ΔE, so we can say: \( 2.5 \, \text{kJ/mol} = P \Delta V \)

Step 3: Relate the Quantity to the Process

Since \( P \Delta V \) represents the work done by the system on its surroundings during the process, the 2.5 kJ/mol difference indicates the amount of work done by the system as it expands and the liquid water transformed into water vapor at constant pressure and temperature. This work includes doing mechanical work on the surroundings, such as displacing the atmosphere or lifting a piston.

Step 4: Conclusion

The 2.5 kJ/mol difference between ΔH and ΔE represents the work done by the system on its surroundings as the water molecules transform from a liquid phase to gaseous phase, at constant pressure and temperature (298 K and 1.0 atm).

Key Concepts

Internal Energy

In thermodynamics, internal energy is a key concept, encompassing the total energy contained within a system. This includes both the kinetic and potential energy of molecules. The kinetic energy arises from the movement of particles, whereas the potential energy is due to the forces acting between the particles.
When discussing processes like the conversion of liquid water to vapor, we often refer to changes in internal energy (\(\Delta E\)). This change occurs because the energy level in a substance shifts as it undergoes a transformation.
A key formula to understand is:\[\Delta E = q - w\]where \(q\) represents the heat added to the system and \(w\) the work done by the system. Internal energy is crucial when discussing state changes, as it provides insight into how energy is distributed within the system.

State Functions

State functions provide a simplified way to describe the behavior of a system. They define the state of a system based on its current condition, irrespective of how it came to that state. Enthalpy (\(H\)) and internal energy (\(E\)) are classic examples of state functions.
These functions are crucial because they allow the calculation of system properties without needing to track the detailed path taken by the system. Instead, only the initial and final states matter.

• Enthalpy (\(H\)): Represents heat released or absorbed at constant pressure.
• Internal Energy (\(E\)): Represents total energy content within the system.

Both help describe energy changes in a system during processes like phase changes, which can often include significant transformations of energy and matter.

Phase Change

A phase change occurs when a substance transforms from one state of matter to another, such as from liquid to gas. During a phase change, energy input or removal doesn't alter the temperature of the substance but instead changes the structure or phase.
For instance, when water vaporizes (\(H_{2}O(l) \rightarrow H_{2}O(g)\)), energy is used to separate the molecules, facilitating the transition from liquid to gas. At constant pressure and temperature, this process involves significant enthalpy changes (\(\Delta H\)) due to the energy required to overcome intermolecular forces.
The difference between enthalpy and internal energy change for such transitions is often work done, represented in calculations by changes in volume (\(P\Delta V\)). Thus, in our case, the 2.5 kJ/mol indicates the work performed on the surroundings as water changes its phase.