Question

A system absorbs \(50 \mathrm{~kJ}\) heat and does \(20 \mathrm{~kJ}\) of work. What is the net change in the internal energy of the system? (a) Increase by \(30 \mathrm{~kJ}\) (b) Decrease by \(30 \mathrm{~kJ}\) (c) Increase by \(70 \mathrm{~kJ}\) (d) Decrease by \(70 \mathrm{~kJ}\)

Short answer

Increase by 30 kJ

Step-by-step solution

Identify the Given Information

The system absorbs heat, which is given as 50 kJ. It also does work, which is given as 20 kJ.

Apply the First Law of Thermodynamics

According to the first law of thermodynamics, the change in internal energy (\(\Delta U\)) of a system is equal to the heat added to the system (\(Q\)) minus the work done by the system (\(W\)). The formula is: \[\Delta U = Q - W\].

Insert the Known Values into the Equation

Inserting the known quantities into the equation provides: \[\Delta U = 50\ \mathrm{kJ} - 20\ \mathrm{kJ}\].

Calculate the Net Change in Internal Energy

Performing the subtraction we get: \[\Delta U = 30\ \mathrm{kJ}\].

Determine the Direction of Change

Because heat was added and work was done by the system, the internal energy increased by 30 kJ. Thus, the net change in internal energy is an increase of 30 kJ.

Key concepts

Internal Energy

In thermodynamics, 'internal energy' is one of the most fundamental concepts. It refers to the total energy contained within a system, which is the sum of all the kinetic and potential energies of its molecules. This energy is closely associated with the temperature of the system as well as its phase (solid, liquid, or gas).

When a substance absorbs heat, its molecules move faster or get farther apart, typically increasing the system's internal energy. Conversely, when a system does work on its surroundings or releases heat, its internal energy tends to decrease. In the context of our exercise, when a system absorbs 50 kJ of heat and does 20 kJ of work, it's like pouring 50 kJ of energy into a container and then using 20 kJ to power a machine. The 'container', which represents the internal energy of the system, ends up with a net gain of 30 kJ.

Thermodynamics

Thermodynamics is a branch of physics that deals with the relationships between heat, work, temperature, and energy. The laws of thermodynamics are the foundation upon which these phenomena can be understood and quantified.

Specifically, the first law of thermodynamics, also known as the law of energy conservation, asserts that energy can neither be created nor destroyed. It can only change forms or be transferred from one system to another. This law is essentially a statement about the conservation of energy, and it provides a balance sheet for energy transactions in any process.

Applying this law in our example, when the system absorbs heat (energy input) and performs work (energy output), the first law helps determine the new energy state or 'balance' of the system. After adding and subtracting energy transactions (heat absorbed and work done), we directly infer the change in the system's internal energy.

Heat and Work

The concepts of 'heat' and 'work' are intrinsically linked to the first law of thermodynamics. Heat (\(Q\)) is a form of energy transfer that occurs between systems due to a temperature difference, while work (\(W\)) is the energy transfer resulting from a force applied over a distance.

In thermodynamic processes, when heat is added to a system, it can increase the system's internal energy, or it can cause the system to perform work, or it can do both. The beauty of the first law is in its simplicity: the increase in internal energy is just the net effect of these two energy transfers.

In the example we're examining, the system absorbs heat and also does work. Since doing work is essentially energy leaving the system, we express this as a subtraction from the heat absorbed. Hence, if a system absorbs 50 kJ of heat and does 20 kJ of work, the internal energy increases by the difference of the two, reinforcing the intuitive notion that you 'keep' what isn't 'spent' – a straightforward yet profound illustration of energy conservation.