Question

Calculate \(\Delta E\) for each of the following cases. a. \(q=+65 \mathrm{~kJ}, w=-22 \mathrm{~kJ}\) b. \(q=+200 . \mathrm{kJ}, w=-73 \mathrm{~kJ}\) c. \(q=-18 \mathrm{~kJ}, w=-40 . \mathrm{kJ}\)

Short answer

In conclusion, the changes in internal energy for each case are: a. \(\Delta E = 43\,\mathrm{kJ}\) b. \(\Delta E = 127\,\mathrm{kJ}\) c. \(\Delta E = -58\,\mathrm{kJ}\)

Step-by-step solution

a. Calculate \(\Delta E\) for \(q=+65\,\mathrm{kJ}\) and \(w=-22\,\mathrm{kJ}\)

First, plug the values of \(q\) and \(w\) into the \(\Delta E\) formula: \[\Delta E = q + w = (+65\,\mathrm{kJ}) + (-22\,\mathrm{kJ})\] Now, add the values: \[\Delta E = 43\,\mathrm{kJ}\]

b. Calculate \(\Delta E\) for \(q=+200\,\mathrm{kJ}\) and \(w=-73\,\mathrm{kJ}\)

First, plug the values of \(q\) and \(w\) into the \(\Delta E\) formula: \[\Delta E = q + w = (+200\,\mathrm{kJ}) + (-73\,\mathrm{kJ})\] Now, add the values: \[\Delta E = 127\,\mathrm{kJ}\]

c. Calculate \(\Delta E\) for \(q=-18\,\mathrm{kJ}\) and \(w=-40\,\mathrm{kJ}\)

First, plug the values of \(q\) and \(w\) into the \(\Delta E\) formula: \[\Delta E = q + w = (-18\,\mathrm{kJ}) + (-40\,\mathrm{kJ})\] Now, add the values: \[\Delta E = -58\,\mathrm{kJ}\] In conclusion, the changes in internal energy for each case are: a. \(\Delta E = 43\,\mathrm{kJ}\) b. \(\Delta E = 127\,\mathrm{kJ}\) c. \(\Delta E = -58\,\mathrm{kJ}\)

Key concepts

Internal Energy

Internal energy is a core concept in thermodynamics that deals with the total energy contained within a system. It's important to understand that this energy includes all forms stored within the system, such as kinetic and potential energies at the microscopic level, comprising the vibrational, rotational, and translational motions of molecules.
In a closed system, internal energy can change when energy is exchanged with the surroundings, usually in the form of heat or work, but it is fundamental to know that energy itself cannot be created or destroyed according to the law of conservation of energy.

• The internal energy of a system can increase if heat is absorbed or work is done on the system.
• Conversely, the internal energy decreases if the system releases heat or does work on its surroundings.

In calculations, internal energy is typically denoted by a change, \(\Delta E\), which symbolizes how the internal energy has shifted from one state to another.

Heat and Work

In thermodynamics, heat and work are two distinct methods by which energy can be transferred between a system and its surroundings. They play a vital role in the calculation of changes to a system's internal energy.
Heat, symbolized by \(q\), involves energy transfer due to temperature differences. When heat is absorbed by the system, it's positive, indicating an energy gain. If heat flows out, it is negative.
Work, represented by \(w\), concerns energy transfer due to mechanical processes, such as pressure-volume work (like expanding or compressing gases).

• Work is positive when the surroundings do work on the system, typically compressing the system.
• It is negative when the system does work on the surroundings which often involves expansion.

These tools provide concrete ways to understand how energy is exchanged, contributing to alterations in the internal energy of systems.

Energy Calculation

Energy calculations in thermodynamics often revolve around the first law of thermodynamics, which highlights the relationship between internal energy, heat, and work. The formula \(\Delta E = q + w\) is commonly used to quantify the change in internal energy.
Let's break it down:

• \(q\) represents the heat absorbed or released. Positive \(q\) shows heat absorption; negative \(q\) shows heat release.
• \(w\) depicts work done on or by the system. Positive \(w\) is work done on the system, whereas negative \(w\) indicates work done by the system.

By adding the values of \(q\) and \(w\), you can determine \(\Delta E\), providing insight into whether the system's internal energy has increased or decreased.
When tackling problems, remember:

• Identify and assign the correct sign to the values of \(q\) and \(w\) based on whether energy is entering or leaving the system.
• Add these values together carefully to find the total change in internal energy for the exact outcome.

This formula is crucial for understanding energy transfers and transformations in thermodynamic processes.