First law of thermodynamics

The first law of thermodynamics states that the change in energy of a closed system equals the sum of heat transferred to it and work done to it

\d U = \db Q + \db W.

This is basically a statement of conservation of energy.

Here $\db$ denotes an inexact differential. This means that there are no functions $Q$ and $W$ of which $\db Q$ and $\db W$ are differentials. There is no measurement of a system I could perform which would tell me it’s “heat” or it’s “work”. $\d U$ is an exact differential, so I could on the other hand measure the energy or temperature.

In other words, $U$ is a state function, while $Q$ and $W$ are not.

Entropy is also a state function. Since there exists some $S(U,V)$ we can write $U(S,V)$ to say

\begin{align*} \d U &= \left.\frac{\partial U}{\partial S}\right|_V \d S + \left.\frac{\partial U}{\partial V}\right|_S \d V \tag{generally} \\ &= \frac 1T \d S - p \d V. \tag{quasistatic} \end{align*}