First Law of Thermodynamics

The first law of thermodynamics is one of the three fundamental laws of thermodynamics. It is derived from the conservation of energy but is stated in a different and more useful manner for thermodynamics in order to include systems where the main method of energy transfer is heat transfer and work.

The first law of thermodynamics: Equation

The first law of thermodynamics was derived in the 19th century by Rudolf Clausius and William Thomson. It states that the total change in the internal energy ΔU of a closed system is equal to the total heat transfer supplied into the system Q minus the total work done by the system W.

Figure 1. First law of thermodynamics. Source: Oğulcan Tezcan, StudySmarter.

First law of thermodynamics: Internal energy (U)

The internal energy of a system on a smaller scale can be considered as the stored sum of the kinetic and potential energy of its atoms and molecules. However, it is more useful to define the internal energy on a larger scale, using the macroscopic quantities of a system, including pressure, temperature, and volume, to study the system’s behaviour.

• Internal energy can be positive when heat is added to the system and/or work is done on the system.

• Internal energy can be negative when heat is removed from the system and/or work is done by the system on its surroundings.

First law of thermodynamics: Heat (Q)

Heat (Q), measured in joules, is the energy that is transferred by molecular motion and collisions due to a temperature difference. When the system is taken as the reference, the heat that enters a system can be considered to be positive, while the heat exiting a system is negative.

First law of thermodynamics: Work (W)

The work (W) of a system, measured in joules, is the energy that is transferred from one system to another or to its surroundings. This is a general form of mechanical work.

• When work is done by the system of reference, the work is defined as negative, as energy is lost from the system of reference and consumed by an external system or the surroundings.
• When work is done on the system of reference, the work is defined as positive, as energy is added to the system of reference and lost from an external system or the surroundings.

Some examples of positive and negative work depending on the chosen system of reference are given in the table below.

First law of thermodynamics in differential form

The differential form of the first law of thermodynamics can be seen below. The differential form of the equation is used to describe in more detail the rate of change of heat and work and, as an extension, the rate of change of a system’s internal energy.

\[\partial U = \partial Q - \partial W\]

In the case of the work done in a hydrostatic system, a system containing fluids, the differential equation can be simplified as shown below, where p is the pressure, and V is the volume of the system. Hence, the first law of thermodynamics can also be written as shown below when the volume of a fluid changes.

\[\partial W = -p\partial V \qquad \partial U = \partial Q + p \partial V\]

The negative sign indicates that the changes in volume are always opposite to the sign of the changes in work. For example, if work is positive, dV would be negative, and vice versa.

First law of thermodynamics examples

The most common application of the first law of thermodynamics is the heat engine, which is used in trains, vehicles, etc.

Other applications include aeroplane engines, refrigerators systems, and heat pumps.

First law of thermodynamics: Thermodynamic systems

There are three types of systems that can be observed in thermodynamics:

• Open systems, which exchange both energy and matter with their surroundings. For example, when boiling water in a pan, energy and matter are transferred from the pan to the surrounding atmosphere as steam.

• Closed systems, which exchange only energy with their surroundings. For example, a hot cup of coffee with a lid on transfers energy from the coffee to the surrounding atmosphere in the form of steam.

• Isolated systems, which are a special case of closed systems that transfer neither energy nor matter to other systems or their surroundings. For example, a perfectly insulated and closed nitrogen tank transfers neither energy nor matter to its surroundings.

How does the first law of thermodynamics apply to gases?

Gases are sensitive to changes in macroscopic quantities such as volume, temperature, and density. For example, when the temperature increases, gases tend to expand due to the increased kinetic energy of the gas molecules. When the temperature decreases, the gases tend to compress. For constant pressure, the formula below can be used, where p is the pressure while Δv is the change in volume. The minus sign indicates that the work is done with respect to the system.

\[W = -p \cdot \Delta V\]

From a thermodynamic perspective, the following apply:

• When a gas expands, the energy is transferred to the system’s surroundings. Work is done by the gas on the surroundings. Here the work is negative (-W) with respect to the system (gas), as energy is released from the system.

• When a gas is compressed, energy is transferred from the surroundings to the gas. Work is done on the gas by the surroundings. Hence the work is positive (+ W) with respect to the system (gas).

• If the work done is being considered with respect to the surroundings, then the sign in the equation becomes positive. Work done becomes positive when the gas is expanded, while the work done is negative when the gas is compressed.