Problem 58 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

The latent heat of vaporization of a liquid at \(500 \mathrm{~K}\) and 1 atm pressure is \(10 \mathrm{kcal} / \mathrm{mol}\). What will be the change in internal energy if 3 moles of the liquid changes to vapour state at the same temperature and pressure? (a) \(27 \mathrm{kcal}\) (b) \(13 \mathrm{kcal}\) (c) \(-27 \mathrm{kcal}\) (d) \(-13\) kcal

Step-by-Step Solution

Answer

The change in internal energy when 3 moles of the liquid changes to vapor state is 30 kcal (none of the given options).

Step 1: Identify the relevant quantities given in the problem

The problem provides us with the latent heat of vaporization of the liquid, which is 10 kcal/mol, and the amount of liquid which is 3 moles.

Step 2: Understanding the concept of latent heat

Latent heat of vaporization is the heat required to convert 1 mole of a substance from the liquid state to the vapor state without a change in temperature. Since pressure and temperature remain constant in this process, the change in internal energy is equal to the heat absorbed.

Step 3: Calculate the total heat absorbed for vaporization

Since the latent heat is given per mole, multiply the latent heat by the number of moles to obtain the total heat absorbed for 3 moles, which is 10 kcal/mol * 3 moles = 30 kcal.

Step 4: Determine the change in internal energy

According to the first law of thermodynamics, the change in internal energy is equal to the heat absorbed when no work is done. Here, no work is done since the pressure and volume are constant; therefore, the change in internal energy is also 30 kcal.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat and other forms of energy. In the context of this exercise, thermodynamics helps us understand how energy in the form of heat, can cause a substance to go from one phase to another, such as liquid to vapor.
Thermodynamics is governed by four fundamental laws, and one of the most crucial concepts we apply here is the behavior of substances when they absorb or release heat without changing temperature – a process known as a phase change. In the given exercise, we observe this during the vaporization of a liquid, where heat is absorbed but the temperature remains constant.
An understanding of thermodynamics is essential for grasping how different energy exchanges lead to various physical changes in matter. It's also central for explaining the latent heat of vaporization, which directly intersects with the principles of thermodynamics.

Chemical Thermodynamics

Chemical thermodynamics is the subset of thermodynamics that deals with energy changes during chemical reactions and phase changes. It is essential in describing the latent heat of vaporization from the molecular perspective.
In the context of the exercise, the chemical thermodynamics approach looks at how the internal energy of the system, which includes the potential and kinetic energy of its molecules, changes as the liquid turns into vapor. The latent heat of vaporization in this sense is a form of energy required to break the intermolecular forces within the substance, allowing it to change from a liquid to a gaseous state at a constant temperature.
This thermodynamic function narrates a tale of molecules gaining sufficient energy to overcome the attractive forces that hold them in a liquid form, which allows students to connect macroscopic heat transfer with microscopic interactions.

First Law of Thermodynamics

The first law of thermodynamics, also known as the law of energy conservation, states that energy cannot be created or destroyed in an isolated system; it can only be transformed or transferred. In the language of the problem at hand, the internal energy change of a system is equal to the heat added to the system minus the work done by the system on its surroundings.

Energy Transfer in Phase Changes

In a phase transition, like vaporization, work is not performed if the process occurs at constant pressure and temperature, so all the heat absorbed (latent heat of vaporization) translates directly into the change in internal energy. This principle allows us to calculate the internal energy change as simply the product of the latent heat and the amount of substance, without worrying about work performed, offering a more straightforward solution to the problem presented.

Internal Energy

Internal energy is a concept rooted in thermodynamics that represents the total energy contained within a system. It encompasses all forms of energy in a system, including both the kinetic energy of particles moving and vibrating, and the potential energy stored within the bonds between these particles.

Internal Energy in Vaporization

The change in internal energy during the vaporization process reflects the energy absorbed by the molecules to transition into a less ordered, more energetic state. In the provided exercise, we calculate the change in internal energy during vaporization by considering the latent heat of vaporization, which is effectively the amount of energy required to facilitate this phase change for a given quantity of substance.
Thus, when the liquid vaporizes, the internal energy increases by the energy associated with this absorbed heat. This explains why, in the exercise, the change in internal energy is a positive value, indicating an increase in the system's total energy as a result of the vaporization.