First Law of Thermodynamics

The First Law of Thermodynamics

The first law of thermodynamics is the law of conservation of energy applied to thermodynamic systems. It states that energy can neither be created nor destroyed; it can only change forms. When a system undergoes a thermodynamic process, the total change in its internal energy equals the heat added to the system minus the work done by the system.

This is the central quantitative statement of the law. Every thermodynamic process must obey it.

Here $\Delta U$ is the change in internal energy of the system, $Q$ is the heat supplied to the system, and $W$ is the work done by the system on its surroundings.

The sign convention is crucial. Heat added to the system is positive ($Q > 0$). Work done by the system is positive ($W > 0$). If the system loses heat, $Q$ is negative. If work is done on the system, $W$ is negative.

The Meaning of Each Term

Internal energy ($U$) is a state function — its value depends only on the current state of the system (pressure, volume, temperature, composition), not on how the system reached that state. For an ideal gas, internal energy depends only on temperature. A change in internal energy $\Delta U$ is path-independent: it is the same for all processes connecting the same two states.

Heat ($Q$) is energy transferred between the system and surroundings due to a temperature difference. Heat is not a property of the system; it is energy in transit. The amount of heat transferred depends on the path taken between states.

Work ($W$) is energy transferred by mechanical means — typically, expansion or compression of the system against external pressure. Like heat, work is path-dependent. For a gas expanding or contracting, the work done by the system is $W = \int P \, dV$, where $P$ is the pressure of the gas.

The First Law for Different Processes

The first law takes specific forms depending on what quantities are held constant during a process.

Isothermal process (constant temperature): For an ideal gas, internal energy depends only on temperature. If temperature does not change, $\Delta U = 0$. The first law becomes $0 = Q - W$, or $Q = W$. All heat added to the system is converted entirely into work done by the system.

Isochoric process (constant volume): When volume is constant, no work is done ($W = 0$). The first law reduces to $\Delta U = Q$. All heat added goes entirely into increasing the internal energy of the system.

Isobaric process (constant pressure): Work done is $W = P \Delta V$. The first law becomes $\Delta U = Q - P \Delta V$.

Adiabatic process (no heat exchange): $Q = 0$, so $\Delta U = -W$. The system does work at the expense of its internal energy, causing its temperature to drop. Conversely, work done on the system increases its internal energy and raises its temperature.

Cyclic Processes

A cyclic process is one that returns the system to its initial state. Since internal energy is a state function, $\Delta U = 0$ for a complete cycle. The first law then gives $0 = Q_{\text{net}} - W_{\text{net}}$, or $Q_{\text{net}} = W_{\text{net}}$.

The net heat absorbed by the system over one complete cycle equals the net work done by the system over that cycle. This is the basis for understanding heat engines.

The First Law as a Statement of Energy Conservation

The first law can be written in differential form for infinitesimal changes:

The notation $\delta Q$ and $\delta W$ (instead of $dQ$ and $dW$) emphasises that heat and work are not exact differentials — they depend on the path. Internal energy $U$, however, is an exact differential: its integral depends only on the endpoints.

Worked Example: Applying the First Law

Consider a gas that absorbs 200 J of heat and does 150 J of work on its surroundings. The change in internal energy is:

The internal energy increases by 50 J.

If instead the gas does 250 J of work while absorbing 200 J of heat:

The internal energy decreases by 50 J — the gas has drawn energy from its own internal store to do the extra work.

Worked Example: Vaporisation of Water (the Textbook's Own Calculation, Eq. 11.3)

The textbook grounds the first law with a real numerical example: what happens, energetically, when 1 g of water turns into steam?

The latent heat of vaporisation of water is $L = 2256\ \text{J/g}$, so converting 1 g of water entirely to steam at its boiling point requires a heat input of $\Delta Q = 2256\ \text{J}$.

This heat does two things: it does work pushing back the atmosphere to make room for the much larger volume of steam, and it increases the internal energy of the system (mainly by pulling water molecules apart against their mutual attraction). …