2.5 Work

Applying the general definition of differential work to an expansion against a constant external pressure \(p_{ex}\) we find for the expansion work

Note: EXPANSION work has to do with:

1. EXTERNAL pressure.

2. Change in volume of the system.

3. The sign is given by convention, work done by the system is negative.

4. In general \(w\) is path dependent.

Examples for expansion work are

• Joule experiment, i.e. free expansion (against \(p_{ex} = 0\)). \(w = 0\), but \(dU = \delta q \neq 0\). Obviously any derivative of energy with respect to volume has the dimension of pressure which motivates the meaning of \(\partial U / \partial V\) to be an ”internal pressure”.

• Expansion against constant pressure:

  \begin{equation*} w = - \int_{V_1}^{V_2} p_{ex} dV = - p_{ex} \Delta V \label{eq:w_pexconst} \end{equation*}

  (2.14)

  (rectangular area in a \(p - V\) diagram).

• Reversible and isothermal expansion:

  \begin{equation*} \delta w = - p_{ex} dV = - p dV \quad \mbox{(internal pressure = external pressure for reversible changes)} \label{eq:w_pex_equilibrium} \end{equation*}

  (2.15)

  \(p\) and \(V\) change, thus a thermal equation of state is needed. As an example we use the perfect gas equation

  \begin{equation*} w = - \int_{V_1}^{V_2} p dV = - n\, R\, T\int_{V_1}^{V_2} \frac{dV}{V} = - n\, R\, T \ln \frac{V_2}{V_1} \label{eq:w_id_gas} \end{equation*}

  (2.16)

Several types of parameter pairs exist whose product represents an energy:

© J. Carstensen (TD Kin I)