B.4.1 First law of thermodynamics (HL only)

Energy Conservation in Thermodynamics

Thermodynamics explores how energy is transferred and transformed, particularly through heat and work.

These concepts are governed by the first law of thermodynamics, which is a specific application of the law of conservation of energy.

The First Law of Thermodynamics: Energy Conservation

Definition

First law of thermodynamics

The first law of thermodynamics states that the total energy of an isolated system is constant.

In other words, energy can be transferred or transformed, but it cannot be created or destroyed.

Note

The first law of thermodynamics is a specific application of the law of conservation of energy to thermodynamic systems.

Internal Energy, Heat, and Work

To understand how energy is conserved in thermodynamics, we need to consider three key components:

Internal Energy ( $U$ ): The total energy contained within a system, including the kinetic and potential energy of its particles.
Heat ( $Q$ ): The energy transferred between a system and its surroundings due to a temperature difference.
Work ( $W$ ): The energy transferred when a force acts over a distance, such as when a gas expands or is compressed.

The First Law of Thermodynamics: Mathematical Formulation

The first law of thermodynamics can be expressed mathematically as:

$Q = Δ U + W$

where:

$Q$ is the heat added to the system.
$Δ U$ is the change in internal energy of the system.
$W$ is the work done by the system.

Note

The sign convention is important:

$Q > 0$ : Heat is added to the system.
$Q < 0$ : Heat is removed from the system.
$W > 0$ : Work is done by the system (e.g., expansion).
$W < 0$ : Work is done on the system (e.g., compression).

How Energy is Conserved

The equation $Q = Δ U + W$ shows that the energy added to a system as heat (Q) is used to either:

Increase the system’s internal energy ( $Δ U$ ).
Perform work ( $W$ ) on the surroundings.

Tip

If no heat is added or removed ( $Q = 0$ ), any work done by the system must come from its internal energy, leading to a decrease in $U$ .

Example

Imagine a gas in a cylinder with a piston. If you heat the gas, it can expand, pushing the piston outward. The energy you supplied as heat is used to increase the gas’s internal energy and to do work on the piston.

Work Done on or by a Gas

In thermodynamics, work is often associated with changes in the volume of a gas.

Work and Volume Changes

When a gas expands or is compressed, work is done.

The work done by a gas during a volume change is given by the formula:

$W = P Δ V$

where:

$W$ is the work done by the gas.
$P$ is the pressure of the gas (assumed constant during the process).
$Δ V$ is the change in volume of the gas.

Note

If the gas expands, $Δ V > 0$ and $W > 0$ (work is done by the gas).

If the gas is compressed, $Δ V < 0$ and $W < 0$ (work is done on the gas).

Calculating Work from Pressure-Volume Diagrams

If the pressure is not constant, the work done can be determined from the area under the curve on a pressure-volume (P-V) diagram.

Illustrating area under the curve of a P-V diagram.

Example

Consider a gas expanding from an initial volume $V_{1}$ to a final volume $V_{2}$ at a constant pressure of $2.0 \times 10^{5} Pa$ .

If the volume increases from $1.0 m^{3}$ to $1.5 m^{3}$ , the work done by the gas is:

$W = P Δ V$

$= 2.0 \times 10^{5} Pa \times (1.5 m^{3} - 1.0 m^{3})$

$= 1.0 \times 10^{5} J$

Tip

When the pressure is not constant, divide the P-V curve into small segments where the pressure is approximately constant. Calculate the work for each segment and sum the results.

Applying the First Law to Thermodynamic Processes

The first law of thermodynamics can be applied to various thermodynamic processes, each with its own characteristics.

Isothermal Process (Constant Temperature)

In an isothermal process, the temperature of the system remains constant ( $Δ U = 0$ for an ideal gas).
Therefore, the heat added to the system is equal to the work done by the system:

$Q = W$

Adiabatic Process (No Heat Exchange)

In an adiabatic process, no heat is exchanged with the surroundings ( $Q = 0$ ).
The work done by the system comes entirely from its internal energy:

$Δ U = - W$

Example

During an adiabatic expansion, a gas does work on its surroundings, causing its internal energy and temperature to decrease.

Isobaric Process (Constant Pressure)

In an isobaric process, the pressure remains constant.
The first law can be applied directly using the following formula to calculate the work done:

$W = P Δ V$

Isochoric Process (Constant Volume)

In an isochoric process, the volume remains constant, so no work is done ( $W = 0$ ).
The change in internal energy is equal to the heat added to the system:

$Q = Δ U$

Common Mistake

Students often confuse the signs of $Q$ and $W$ .

Remember that positive $Q$ means heat is added to the system, while positive $W$ means work is done by the system.

P-V diagram showing possible thermodynamic processes.

Reflection

Theory of Knowledge

How does the first law of thermodynamics relate to the broader principle of conservation of energy in physics?
Can you think of other systems where energy conservation plays a critical role?

The first law provides a framework for predicting how energy is transferred and transformed in these systems.

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