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 = Ef – Ei) 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.
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.