# 19.4: Energy is a State Function

- Page ID
- 13702

## Work and heat are not state functions

The change in internal energy of a system is the sum of \(w\) and \(q\), which is a state function.

The realization that work and heat are both forms of energy transfer undergoes quite an extension by saying that internal energy is a state function. It means that although heat and work can be produced and destroyed (and transformed into each other), energy is conserved. This allows us to do some serious bookkeeping! We can write the law as:

\[\Delta U = w + q \nonumber \]

But the (important!) bit about the state function is better represented if we talk about small changes of the energy:

\[dU = \delta w +\delta q \nonumber \]

We write a straight Latin \(d\) for \(U\) to indicate when the change in a state function, where as the changes in work and heat are path-dependent. This is indicated by the 'crooked' \(\delta\). We can represent changes as integrals, but only for \(U\) can we say that regardless of path we get \(\Delta U = U_2-U_1\) if we go from state one to state two. (I.e. it only depends on the end points, not the path).

Notice that when we write \(dU\) or \(\delta q\), we always mean infinitesimally small changes, i.e. we are implicitly taking a limit for the change approaching zero. To arrive at a macroscopic difference like \(\Delta U\) or a macroscopic (finite) amount of heat \(q\) or work \(w\) **we need to integrate**.

We will now invoke the **first law of thermodynamics**:

- \(dU=\delta q+\delta w\)
- \(\oint{dU}=0\)
- Internal energy is conserved

These are all ways of saying that internal energy is a state function.