Question

The increase in internal energy of the system is 100 \(\mathrm{J}\) when \(300 \mathrm{~J}\) of heat is supplied to it. What is the amount of work done by the system (a) \(-200 \mathrm{~J}\) (b) \(+200 \mathrm{~J}\) (c) \(-300 \mathrm{~J}\) (d) \(-400 \mathrm{~J}\)

Short answer

(b) +200 J

Step-by-step solution

Understand the Problem

We need to find the amount of work done by the system when the internal energy change and the heat supplied to the system are given.

Know the Formula

The formula used to find the work done by the system when heat is added is the First Law of Thermodynamics: \( \Delta U = Q - W \), 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.

Substitute the Values

Given that \( \Delta U = 100 \, \mathrm{J} \) and \( Q = 300 \, \mathrm{J} \), substitute these into the formula: \[ 100 = 300 - W \].

Solve for Work Done

Rearrange the equation to solve for \( W \): \[ W = 300 - 100 \] \[ W = 200 \, \mathrm{J} \].

Consider the Sign of Work

The calculated work is \( 200 \, \mathrm{J} \). Since energy is used to do work, it must be counted as positive work done by the system. Therefore, the answer is \(+200 \, \mathrm{J}\).

Key concepts

Change in Internal Energy

When studying thermodynamics, one key concept is the change in internal energy, denoted as \( \Delta U \). Internal energy refers to the total energy contained within a system, arising from the kinetic and potential energies of its particles.

Internal energy can change in response to energy added or removed from the system. This is governed by the First Law of Thermodynamics, which is expressed as \( \Delta U = Q - W \). Here, \( Q \) is the heat supplied to the system, and \( W \) is the work done by the system.

The change in internal energy can be:

• Positive when the system gains energy.
• Negative when the system loses energy.

Understanding these changes is crucial for analyzing how a system's properties evolve under various conditions.

Heat Supplied

Heat is energy transferred due to a temperature difference between a system and its surroundings. In thermodynamics, it is a key factor affecting a system's energy balance.

When heat is supplied to a system, it can cause changes in both the internal energy and the work done. This relationship is captured by the First Law of Thermodynamics. For example, when heat is added to a system,

• some of it increases the internal energy, which might raise the system's temperature.
• some might do work, such as expanding a gas.

In the exercise, 300 J of heat was supplied, part of which increased the internal energy by 100 J while the rest was used for work. The balance between these factors is crucial in understanding the system's thermodynamic processes.

Work Done by the System

In thermodynamics, work done by the system is the energy transferred from the system to its surroundings, often involving a force causing movement. It is denoted as \( W \) in equations.

When a system does work, it uses its internal energy. For example, imagine a gas expanding in a piston; the gas does work by pushing the piston outward, which decreases its internal energy. The formula \( \Delta U = Q - W \) shows that work done is subtracted from the heat supplied.

In this exercise:

• The change in internal energy was 100 J after 300 J of heat was supplied.
• The work done by the system was calculated as 200 J, meaning energy was used positively to perform work.

It's important to understand that the sign of work indicates the direction of energy transfer. Positive work indicates energy going out of the system, as in this case where the system did 200 J of work on its surroundings.