Question

A system receives \(425 \mathrm{~J}\) of heat from and delivers \(425 \mathrm{~J}\) of work to its surroundings. What is the change in internal energy of the system (in \(J\) )?

Short answer

The change in internal energy is 0 J.

Step-by-step solution

- Understand the First Law of Thermodynamics

The first law of thermodynamics states that: \[ \Delta U = Q - W \]Here, \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.

- Identify Given Values

From the problem, the heat added to the system \( Q \) is 425 J and the work done by the system \( W \) is 425 J.

- Substitute Values into the Equation

Substitute the given values into the first law of thermodynamics equation:\[ \Delta U = 425\, \mathrm{J} - 425\, \mathrm{J} \]

- Calculate the Change in Internal Energy

Perform the subtraction to find the change in internal energy:\[ \Delta U = 0\, \mathrm{J} \]

Key concepts

Internal energy

The concept of internal energy is fundamental in understanding the first law of thermodynamics. Internal energy, often represented as \( U \), is the total energy contained within a system. This includes the kinetic energy of all particles in the system and the potential energy arising from interactions between these particles. Internal energy can change through heat transfer or work done on or by the system.

To identify changes in internal energy, we look at two things:

• Heat added or removed from the system (\( Q \))
• Work done by or on the system (\( W \))

When heat is added to a system, its internal energy increases. Conversely, when work is done by the system on its surroundings, its internal energy decreases. The first law of thermodynamics connects these effects directly. Let's explore how heat transfer affects internal energy in more detail.

Heat transfer

Heat transfer is a process where energy is transferred to or from a system due to a temperature difference. In thermodynamics, heat transfer is represented by the letter \( Q \). When quoting the first law of thermodynamics, if heat is added to the system, \( Q \) is positive, increasing the total internal energy of the system.

There are three primary modes of heat transfer:

• Conduction: Direct transfer of heat between substances in contact.
• Convection: Transfer of heat by the movement of fluids (liquids or gases)
• Radiation: Transfer of energy through electromagnetic waves.

In the given exercise, the system receives 425 J of heat. This means heat is being transferred into the system, indicating that \( Q = 425 \) J. Now, let's see how the work done influences the internal energy of the system.

Work done

Work done, denoted as \( W \), is another critical component of the first law of thermodynamics. It refers to the energy transferred when a force is applied over a distance. In the context of our exercise, the system does 425 J of work on its surroundings, making \( W \) positive.

According to the first law of thermodynamics, work done by the system decreases its internal energy because energy is leaving the system. In the exercise: \( W = 425 \) J. When work is done on the system, the internal energy increases, and we consider \( W \) as negative. In this particular problem, both the heat added and work done are equal at 425 J each.

When we apply the first law equation \( \Delta U = Q - W \), we substitute the known values to find the change in internal energy: \( \Delta U = 425 \text{ J} - 425 \text{ J} = 0 \text{ J} \). This means there is no net change in the internal energy of the system. The heat energy received is entirely used to do work, keeping the internal energy unchanged.