2 The first law 2.1 Thermometers 2.3 Specific heats

2.2 The first law

(September 1, 2023)

The change in energy is determine by the heat added to the system ${\mathchar 22\mskip-12.0mu d}Q$ plus the work done on the system by you, ${\mathchar 22\mskip-12.0mu d}W$ :

dU={\mathchar 22\mskip-12.0mu d}Q+{\mathchar 22\mskip-12.0mu d}W

(2.2)

The work done by the system on you (or the work you get out) is ${\mathchar 22\mskip-12.0mu d}W_{\rm out}=-{\mathchar 22\mskip-12.0mu d}W$ . So

dU={\mathchar 22\mskip-12.0mu d}Q-{\mathchar 22\mskip-12.0mu d}W_{\rm out}

(2.3)

For a simple homogeneous substance

{\mathchar 22\mskip-12.0mu d}W_{\rm out}=p(T,V){\rm d}V\qquad W_{\rm out}=\int% _{i}^{f}p(T,V)\,{\rm d}V

(2.4)

We put a bar as in ${\mathchar 22\mskip-12.0mu d}W$ because ${\mathchar 22\mskip-12.0mu d}W$ is not the change in a function called $W$ . Rather it is a small amount of work. The amount of work done depends on the path taken (see lecture notes). Similarly ${\mathchar 22\mskip-12.0mu d}Q$ is an amount of heat; it is not the change in a function which we call $Q$ . The amount of heat added or removed from the system depends on the path.

To work with the first law we generally need to specify two functions. The first is the pressure³³ 3 In general the pressure and energy are functions the number of particles $N$ , but $N$ is considered to be a constant and is not notated: $p(T,V)\equiv p(T,V,N)$ $p(T,V)$ and the second is the energy $U(T,V)$

	$\displaystyle P(T,V)\equiv$	the pressure as a function of temperature and volume		(2.5)
	$\displaystyle U(T,V)\equiv$	the energy as a function of temperature and volume		(2.6)

Often the relation pressure-temperature-volume relation is inverted expressing the volume in terms of temperature and pressure

V(T,P)\equiv\mbox{the volume as a function of temperature an pressure}

(2.7)