The increase in internal energy of the system is the sum of heat and work input into the system.

Differential form of first law of thermodynamics:

$$dE = \delta Q + \delta W$$

Integral form of first law of thermodynamics:

$$\Delta E = Q + W$$

,where Q is the net heat input, W is the net work output (mechanical, electrical or others), and ΔE is the change in internal energy (i.e., ΔE = E_{f} – E_{i}) of the system.

– This is really nothing but a definition of “internal energy”

– Nevertheless, it expresses the principle of energy conservation because it says that the energy that enters (or leaves) a system becomes (or comes from) internal energy in the system: the energy is not destroyed or created.

In systems where the work is PV work, the first law can be explicitly written as,

$$dE = \delta Q – P dV$$

If this process is reversible, the heat input is given by TdS, and so the first law can also be written as,

$$dE = T \, dS – P \, dV$$

Note: S is entropy, more about this will be talked about later.

**Alternative sign convention**

Sometimes you see the first law written with a “minus” sign,

$$\Delta E = Q – W$$

This is the same law, but W is not defined to be work done by the system, instead of on the system.

Note: This form is often used for convenience by engineers, since they are mostly interested in the work done by the system.

**Isolated system**

For an isolated system, i.e., one that does not interact with its surroundings, Q = W = 0, and so ΔE must be zero. In other words, the internal energy of an isolated system is constant.