Question

A system releases 255 cal of heat to the surroundings and delivers 428 cal of work. What is the change in internal energy of the system (in cal)?

Short answer

The change in internal energy is -683 cal.

Step-by-step solution

- Understand the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy (\textdelta U) of a system is equal to the heat added to the system (q) minus the work done by the system (w). Mathematically, it is expressed as \(\textdelta U = q - w\).

- Identify Given Values

The problem provides the following values: \( q = -255 \text{ cal}\) (since the system releases heat) and \( w = 428 \text{ cal}\) (since the system does work on the surroundings).

- Substitute Values into the Equation

Substitute the given values into the first law of thermodynamics equation: \(\textdelta U = q - w = -255 \text{ cal} - 428 \text{ cal}\).

- Calculate the Change in Internal Energy

Perform the calculation to find the change in internal energy: \(\textdelta U = -255 \text{ cal} - 428 \text{ cal} = -683 \text{ cal}\).

Key concepts

Internal Energy Change

The concept of internal energy change is rooted in the First Law of Thermodynamics. This law tells us that the internal energy of a system can be altered in two main ways: by adding or removing heat, and by doing work or having work done on the system. In our example exercise, we see the system releasing heat and also doing work. When we calculate internal energy change, we use the formula: \[\textdelta U = q - w \] Here, \(\textdelta U\) represents the change in internal energy, \(q\) stands for heat transfer, and \(w\) is the work done. Both heat release and work done affect the internal energy levels of the system.

• Heat released by the system decreases its internal energy.
• Work done by the system also decreases its internal energy.

This is why, in our exercise, by combining the amounts of heat released and work done, we arrive at the internal energy change.

Heat Transfer

Heat transfer is a crucial concept in thermodynamics. When a system interacts with its surroundings, it can either absorb heat or release heat. In our exercise, the system releases heat, given by \(q = -255 \text{ cal}\). The negative sign indicates that heat is being lost by the system. When heat is added to a system, its energy increases; conversely, when heat is released, the system's energy decreases. Think of heat transfer like adding or removing energy from a storage tank:

• Adding heat = increasing the energy in the tank
• Releasing heat = decreasing the energy in the tank

This fundamental principle helps you understand all other thermodynamic processes. Just remember, energy is always moving, and knowing the direction of heat flow is key.

Work Done

In the context of the First Law of Thermodynamics, work can be thought of as energy transferred by a system to its surroundings. The amount of work done by the system is represented by \(w\). In our example, the system does \(428 \text{ cal}\) of work. When a system does work, it expends energy, thus reducing its internal energy. This is explained by the term 'work done' in our formula: \[\textdelta U = q - w \] Note that if the system had work done on it, it would gain energy, thus altering the signs in our equation. Here's a simplified way to remember:

• System doing work = internal energy decreases
• Work done on system = internal energy increases

Always keep track of the signs as they indicate the direction of energy flow.