Question

If a system does \(45 \mathrm{~J}\) of work and receives \(28 \mathrm{~J}\) of heat, what is the value of \(\Delta E\) for this change?

Short answer

\(\Delta E = -17 \mathrm{~J}\). The system lost 17 joules of internal energy.

Step-by-step solution

Identify the First Law of Thermodynamics

The First Law of Thermodynamics states that the change in internal energy of a system, \(\Delta E\), is equal to the heat added to the system, \(Q\), minus the work done by the system, \(W\): \[\Delta E = Q - W.\]

Substitute the Given Values

Subtract the work done by the system from the heat received by the system using the given values. Let's denote the work done by the system as positive and the heat received as positive as well: \[\Delta E = 28\mathrm{~J} - 45\mathrm{~J}.\]

Calculate the Change in Internal Energy

Carry out the subtraction to find the change in internal energy of the system: \[\Delta E = 28\mathrm{~J} - 45\mathrm{~J} = -17\mathrm{~J}.\]

Key concepts

Understanding Internal Energy in Thermodynamics

In the vast field of thermodynamics, understanding the concept of internal energy is fundamental. Imagine internal energy as the total energy stored within a system. This energy comprises the kinetic energy due to the random motions of the system's particles and the potential energy resulting from the intermolecular forces between those particles.

When we discuss changes in internal energy, denoted as \( \Delta E \), we're referring to the difference between the initial and final energy states of the system. This change can be influenced by transferring energy in two primary ways: heat transfer and doing work on or by the system. In the context of our exercise, the negative result for \(-17 \text{J}\) indicates that the system lost energy, likely leading to a decrease in the temperature or a shift in the molecular structure that reduced potential energy.

Work Done By the System Explained

The work done by the system plays a pivotal role in thermodynamics and is crucial in comprehending energy transformations. When a system does work on its surroundings, it expends energy, manifesting as pressure-volume work among other forms.

If you picture a piston compressing a gas, work is conducted by the system (the gas) pushing against the force needed to compress it. In our exercise, the system does work amounting to \(+45 \text{J}\). This signifies an outflow of energy, thereby decreasing the internal energy of the system. It is vital to keep track of the signs; work done by the system is considered positive in this scenario. However, some problems may define it with a negative sign, so always keep an eye on your conventions!

The Role of Heat Transfer

The transfer of energy via heat transfer is another cornerstone of thermodynamics. It occurs without direct contact or motion, driven by a temperature difference between the system and its surroundings. Heat can move in or out of a system in various ways, such as conduction, convection, or radiation.

In the provided exercise, the system receives \(+28 \text{J}\) of heat. This input of energy increases the system's internal energy. Once more, sign conventions are key: positive values denote heat entering the system, while negative values imply heat loss. Understanding heat transfer not only illuminates how energy moves but also aids in predicting how a system's temperature or phase might change.

Solving Thermodynamics Problems

Tackling thermodynamics problems often involves a blend of theoretical concepts and practical problem-solving skills. The First Law of Thermodynamics, which equates the change in internal energy to the net heat transfer minus the work done by the system, acts as a guiding equation.

Best practices for solving such problems include clearly identifying given data, applying the correct sign conventions, and methodically using the fundamental equations. By breaking down the problem into steps, as seen in the solution we're discussing, and maintaining a meticulous approach, students can avoid common errors and develop a thorough understanding of how energy transformations govern physical systems.