Question

A system releases \(125 \mathrm{kJ}\) of heat while \(104 \mathrm{kJ}\) of work is done on it. Calculate \(\Delta E\).

Short answer

The change in internal energy, ΔE, is \(-229 \mathrm{kJ}\).

Step-by-step solution

Write down the first law of thermodynamics equation

The first law of thermodynamics equation is given by: \[\Delta E= q - w\] where ΔE is the change in internal energy, q is the heat added to the system, and w is the work done by the system.

Plug in the given values

We are given the values for q and w in the problem: q = -125 kJ and w = 104 kJ. Substitute these values into the equation: \[\Delta E = (-125 \mathrm{kJ}) - (104 \mathrm{kJ})\]

Calculate ΔE

Now, perform the calculation: \[\Delta E = -125 \mathrm{kJ} - 104 \mathrm{kJ} = -229 \mathrm{kJ}\] So the change in internal energy, ΔE, is -229 kJ.

Key concepts

Change in Internal Energy

In thermodynamics, the internal energy of a system is a measure of its total energy—encompassing both kinetic energy, due to the movement of particles, and potential energy, which is stored within the system. When we refer to the 'change in internal energy', symbolized as \(\Delta E\), we're talking about the difference between the system's initial and final internal energy states.

The internal energy of a system can change in two fundamental ways: through heat transfer and the work performed on or by the system. It's key to remember that internal energy is a state function, meaning its change is independent of the path taken; it only depends on the initial and final states of the system.

In the given exercise, after inputting the specified heat release and work done on the system into the first law of thermodynamics equation, the resulting negative value of \(\Delta E\) indicates that the system has lost energy overall. This could suggest cooling or compression occurred, wherein the system's particles are slowing down or are being pressed into a smaller volume, both of which decrease internal energy.

Heat Transfer

Heat transfer is a fundamental concept in thermodynamics that describes the movement of thermal energy from one place to another due to a temperature difference. It's measured in Joules in the International System of Units, signifying the amount of energy being transferred.

There are three primary modes of heat transfer: conduction, through direct contact; convection, where heat is carried by moving fluids; and radiation, which involves heat moving via electromagnetic waves. In our exercise scenario, the system releases 125 kJ of heat, implying that the energy is being transferred out of our system and into the surrounding environment.

Heat transfer plays a crucial role in determining the change in a system's internal energy. It's important to bear in mind that, conventionally, positive heat transfer \((q)\), symbolizes heat being absorbed by the system, whereas a negative \(q\) value suggests the system is releasing heat, which is exactly the case outlined in the exercise.

Work Done on System

Within the realm of thermodynamics, work is a form of energy transfer where force is applied over a distance. The type of work we often discuss in thermodynamics concerns the expansion or compression of gases within a system—this is frequently visualized as a piston moving within a cylinder.

In the provided exercise, work done on the system is denoted as positive 104 kJ. This is a direct application of the convention that if work is done on the system, like compressing a gas, it is assigned a positive value. Conversely, if the system does work on the surroundings, such as a gas expanding and pushing a piston outwards, the work value would be negative.

This positive work contributes to an increase in the system's internal energy because energy is being input into the system. However, due to heat loss being greater than the work done on the system, the system's internal energy decreases in our problem, as clearly represented by the calculated negative change in internal energy.