Question

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

Short answer

The 2.5 kJ/mol quantity represents the work done by the system due to the expansion of the gas during the process, which is the PΔV term. This means that the process involves the conversion of 2.5 kJ/mol of energy into work against the external pressure during the transformation of liquid water into water vapor.

Step-by-step solution

Recall the formula relating ΔH, ΔE, and work done by the system

The relationship between the change in enthalpy (ΔH) and the change in internal energy (ΔE) of a system is given by the following formula: ΔH = ΔE + PΔV where P is the pressure and ΔV is the change in volume during the process.

Rearrange the formula

We are given that ΔH is more positive than ΔE by 2.5 kJ/mol. We can represent this using the above equation and solve for PΔV: ΔH - ΔE = PΔV

Plug in the given values

We are given that ΔH - ΔE is 2.5 kJ/mol. Therefore, we have: 2.5 kJ/mol = PΔV

Interpret the result

From the last equation, we can see that the 2.5 kJ/mol quantity represents the work done by the system due to the expansion of the gas during the process, which is PΔV term. Thus, the process involves the conversion of 2.5 kJ/mol of energy into work against the external pressure during the transformation of liquid water into water vapor.

Key concepts

Understanding Enthalpy

Enthalpy, often denoted as \( \Delta H \), is a concept that helps us understand the heat change associated with a process occurring at constant pressure. When we talk about changes in enthalpy, we are looking at the heat absorbed or released during a chemical or physical transformation.
For the transition of liquid water to water vapor (as seen in our example with \( \mathrm{H}_{2}\mathrm{O}(l) \rightarrow \mathrm{H}_{2}\mathrm{O}(g) \)), the heat absorbed is what causes the liquid to overcome intermolecular forces and expand into a gas.

• The enthalpy change is a key indicator of the energy required for this phase change.
• In our exercise, \( \Delta H \) that is more positive suggests that energy is taken in by the system, particularly in the form of heat.
• Since the transformation involves the expansion against atmospheric pressure, there is a direct interaction with the concept of work done by the system.

Understanding enthalpy in this context helps us analyze energy exchanges in various processes, making it fundamental in thermodynamics.

Exploring Internal Energy

Internal energy, denoted as \( \Delta E \), represents the total energy contained within a system. It includes the kinetic energy of molecules moving and vibrating, as well as potential energy from intermolecular forces.
In our process of converting liquid water to vapor, \( \Delta E \) gives insight into the core energy changes, minus the influence of work done by external factors.

• This energy change is strictly about what's happening internally within the molecules.
• The discrepancy between \( \Delta H \) and \( \Delta E \) in the exercise arises because not all energy goes into changing the internal state. Some energy is employed against external pressure to expand the gas.
• In essence, \( \Delta E \) reflects the pure energy change associated with the rearrangement and movement of molecules.

Gaining a grasp of internal energy helps elucidate the fundamental shifts happening in a system at the molecular level.

Work Done by the System

Work done by the system, often represented in thermodynamic equations as \( P\Delta V \), describes the energy transfer that occurs when a system expands or contracts against external pressure. It's an interesting aspect of thermodynamics since it involves energy being converted to do work.
In the given process, water vaporizing creates an expansion that performs work against the atmospheric pressure.

• This work is quantified by the difference \( \Delta H - \Delta E = P\Delta V \), which in this exercise equals 2.5 kJ/mol.
• The 2.5 kJ/mol directly corresponds to the energy turned into work during the phase transition.
• Understanding this term gives clarity on how energy isn't just contained in the substances but also in the action they cause in moving the surroundings.

Thus, work done by the system is a pivotal concept when analyzing processes that involve changes in phase, volume, and other states.