Question

A gas expands and does \(P V\) work on the surroundings equal to \(325 \mathrm{~J}\). At the same time, it absorbs \(127 \mathrm{~J}\) of heat from the surroundings. Calculate the change in energy of the gas.

Short answer

The change in energy of the gas is \(-198\, \mathrm{J}\).

Step-by-step solution

Understand the Concepts

The change in energy of a system can be calculated using the first law of thermodynamics, which states \( \Delta U = Q - W \), where \( \Delta U \) is the change in internal energy, \( Q \) is the heat added to the system, and \( W \) is the work done by the system.

Identify the Given Values

From the problem, you know that the heat absorbed by the gas, \( Q \), is \( 127 \, \mathrm{J} \), and the work done by the gas, \( W \), is given as \( 325 \, \mathrm{J} \).

Apply the Formula

Use the formula \( \Delta U = Q - W \). Place the given values into the formula: \( \Delta U = 127 \, \mathrm{J} - 325 \, \mathrm{J} \).

Calculate the Change in Energy

Perform the subtraction: \( 127 \, \mathrm{J} - 325 \, \mathrm{J} = -198 \, \mathrm{J} \).

Interpret the Result

The result \( -198 \, \mathrm{J} \) indicates the energy of the gas decreases by \( 198 \, \mathrm{J} \). It loses more energy as work than it gains from the absorbed heat.

Key concepts

Internal Energy Change

When discussing internal energy change, we are delving into the heart of thermodynamics. The internal energy of a system is the total energy contained within it due to its molecular motion and interactions. Changing this energy is central to the First Law of Thermodynamics, which is expressed by the equation \( \Delta U = Q - W \). Here, \( \Delta U \) stands for the change in internal energy. By knowing how much heat \( Q \) is absorbed by or added to the system and the work \( W \) done by the system, we can determine this change.
In our exercise, the change is calculated by subtracting the work done by the gas from the heat it absorbs: \( \Delta U = 127 \, \mathrm{J} - 325 \, \mathrm{J} = -198 \, \mathrm{J} \). This negative change indicates that the system loses more energy through work than it gains from absorption.

• Internal energy is cumulative, covering both kinetic and potential energy within the system.
• Energy changes reveal system behavior and interactions.

Heat Transfer

Heat transfer defines the flow of thermal energy from one body or system to another, often due to temperature differences. In our scenario, the gas absorbs heat from its surroundings, quantified by \( Q \). This absorbed heat equals \( 127 \, \mathrm{J} \), energizing the system.
There are three main types of heat transfer: conduction, convection, and radiation. In the context of gas expansion, convection is most relevant, as it involves the bulk movement of gas molecules carrying energy.
Understanding heat transfer helps in identifying how energy is gained or lost, purveying insights into system dynamics:

• Absorbing heat often increases the system's internal energy.
• The sign and magnitude of \( Q \) dictate whether heat is gained or lost.

Work Done by Gas

Work done by a gas involves energy transfer when a gas expands or contracts against an external pressure, a notion neatly captured by the First Law of Thermodynamics as \( W \). When a gas expands, it performs work by pushing against surrounding air or a piston, which consumes internal energy.
In the given task, the gas does \( 325 \, \mathrm{J} \) of work. This significant amount of work leads to a net loss in internal energy unless offset by sufficient heat absorption.
Key points about work done by a gas:

• Expansive work is energy used to overcome external pressures.
• More work implies a greater reduction in internal energy when not counterbalanced by heating.

Exploring how gases perform work unveils the connection between thermodynamic processes and energy conversions in physical systems.