Question

The internal energy of a gas decreases by \(1.65 \mathrm{kJ}\) when it transfers \(1.87 \mathrm{kJ}\) of energy in the form of heat to the surroundings. (a) Calculate the work done by the gas on the surroundings. (b) Does the volume of gas increase or decrease?

Short answer

(a) The work done by the gas on the surroundings is \(0.22 \mathrm{kJ}\). (b) The volume of the gas increases.

Step-by-step solution

Identify Given Values

We are given that the internal energy change, \( \Delta U \), is \(-1.65 \mathrm{kJ}\) and the heat transferred, \( q \), is \(-1.87 \mathrm{kJ}\). The negative signs indicate that energy is leaving the system in both cases.

Understand First Law of Thermodynamics

The first law of thermodynamics states that \( \Delta U = q + w \), where \( \Delta U \) is the change in internal energy, \( q \) is the heat added to the system, and \( w \) is the work done on the system. In this problem, \( w \) will be the work done by the system, and we'll need to solve for \( w \).

Set Up the Equation

Substitute the given values into the first law equation: \( -1.65 \mathrm{kJ} = -1.87 \mathrm{kJ} + w \).

Solve for Work Done

Rearrange the equation to solve for \( w \): \( w = -1.65 \mathrm{kJ} + 1.87 \mathrm{kJ} \). Calculate it to find \( w = 0.22 \mathrm{kJ} \).

Interpret the Result for Work Done

Since the work done \( w \) is positive, it means the gas does \( 0.22 \mathrm{kJ} \) of work on the surroundings.

Determine Change in Volume

In thermodynamics, if the gas does work on its surroundings, it generally means the gas expands. Therefore, the volume of the gas increases.

Key concepts

First Law of Thermodynamics

The First Law of Thermodynamics is like the ultimate accountant of energy. It keeps track of the energy entering and leaving a system. This law can be expressed in a simple equation: 1. \( \Delta U = q + w \) - \( \Delta U \): Change in internal energy of the system - \( q \): Heat added to the system - \( w \): Work done on the system or by the systemIn our example, energy is flowing out, proven by the negative signs for both \( \Delta U \) and \( q \). This indicates a release rather than an uptake. This law is crucial because it shows that energy cannot be created or destroyed, only converted from one form to another. This conversion is central to understanding how systems, like gases in a container, interact with their environment.

Internal Energy

Internal energy is the total energy contained within a system due to its molecular movements and interactions. - **Components of Internal Energy** *Kinetic Energy:* Energy due to molecular movements *Potential Energy:* Energy due to the interactions or configurations of moleculesWhen we say a system's internal energy changes, as it does in this exercise (\( -1.65 \mathrm{kJ} \)), it means that there's been a net loss or gain in energy associated with these movements and interactions. This change is at the heart of how energy transfers affect the physical state of the gas, including things like temperature and pressure.

Heat Transfer

Heat transfer is the movement of energy from one place to another due to temperature differences. - **Mechanism of Heat Transfer** - Conduction - Convection - RadiationIn this exercise, the gas releases \( 1.87 \mathrm{kJ} \) of heat. The direction of this heat tells us that the gas is cooling down, losing energy to its surroundings. This can affect how the gas behaves, often causing it to contract or expand depending on the situation and constraints, such as whether it is in a closed or open container.

Work Done

Work done by a system is the energy it utilizes to move or do something external to itself.- **Understanding Work** - If the system does work on the surroundings, energy is used and work is considered positive. - Conversely, if the surroundings do work on the system, work is negative.In our step-by-step solution, we calculated the work done by the gas to be positive \( 0.22 \mathrm{kJ} \), signaling that energy was expended by the gas to influence its surroundings. This often means in practical terms that the gas expands, as energy is needed to push against the boundaries, thereby increasing the gas's volume. Understanding this process gives insight into gas dynamics in thermal physics.