First law of thermodynamics

A system can gain or lose energy through heating and through work. The first law of thermodynamics applies conservation of energy to these transfers: energy transferred to a system by heating is accounted for by the change in its internal energy and the work it does.

Here, $$Q$$ is the energy transferred to the system by heating, $$\Delta U$$ is the change in its internal energy, and $$W$$ is the work done by the system. All three quantities are measured in joules ($$\mathrm{J}$$).

Rearranging gives $$\Delta U = Q - W$$. Heating into the system increases its internal energy, while work done by the system transfers energy out. The work therefore reduces the increase in internal energy that the heating would otherwise produce.

First law: Sign convention

The signs show the direction of each transfer or change:

• $$Q > 0$$: energy is transferred into the system by heating; $$Q < 0$$: energy is transferred out by heating.
• $$W > 0$$: work is done by the system; $$W < 0$$: work is done on the system.
• $$\Delta U > 0$$: internal energy increases; $$\Delta U < 0$$: internal energy decreases.

Work done on the system has a negative value of $$W$$. Subtracting this negative value in $$\Delta U = Q - W$$ shows that the work increases the system’s internal energy.

First law: Using the equation

The equation can be rearranged to find any one of the three quantities when the other two are known. For example, $$W = Q - \Delta U$$. Retain the signs of the known quantities when substituting them.

Suppose $$100,\mathrm{J}$$ is transferred to a system by heating and $$30,\mathrm{J}$$ of work is done on it. Then $$Q = 100,\mathrm{J}$$ and $$W = -30,\mathrm{J}$$:

The positive result means that the system’s internal energy increases by $$130,\mathrm{J}$$.