Topic: Generalised forces

Recall

List all the extensive variables you know together with their corresponding, conjugate intensive variables?

How do you calculate the work done, when an extensive variable reversibly changes it's value?

Why can we only calculate the change in work like quantity when the state of the system changes reversibly?

Are intensive variables scalar or vector quantities?

Are extensive variables scalar or vector quantities?

Notes

Intensive quantities as generalised forces

Classical thermodynamics is a phenomenological theory that works by exploiting an analogy with Newtonian physics. Extensive quantities are thought of as generalised coordinates, while intensive quantities are thought of as "forces" that act on these generalised coordinates. We thus write the work done in changing any general extensive quantity, E, as:

<aside> 💡 $\Delta w = \int_e^{e+\Delta e} \mathbf{I}(E,\{\Gamma\}) \textrm{d}E$

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In the expression the integrand is the value taken by the intensive quantity that is conjugate to E. The value of this quantity is determined using the equation of state, which is a function of the thermodynamic variables. There are a number of important things to note with regards to this expression

• Every extensive quantity has a corresponding (conjugate) intensive quantity and these two variables appear together when calculating the change in the work-like quantity using the integral above. For some intensive quantities, e.g. pressure it is immediately obvious what the conjugate variable should be - volume. By contrast, for intensive quantities such as temperature or extensive quantities such as the number of atoms what the conjugate variable should be is far from obvious. In fact, we have to introduce new extensive variables in these cases - entropy and the chemical potential.
• Intensive quantities should be thought of as vectors as they have both magnitude and direction. We will see in subsequent lectures that intensive quantities are derivatives and you should by now know that the derivative of a scalar-valued function is always a vector. In thermodynamics, the sign of the intensive quantity tells us what is doing the work and what is being worked upon. So, for example, the pressure is a force applied by the external world to a gas. As such when a gas expands it does work against this force. In other words, the system (the gas) does work on the rest of the world.
• The equation above is only valid when the extensive variable's value is changed reversibly. This is hugely important - when we use this expression we are assuming that our mathematical model provides a complete description of the phenomenon under study.
• When the equation above is used to calculate the work done during a transition the only thermodynamic variables that change during the transition are the extensive variable, E, and its conjugate intensive variable. All the other thermodynamic variables are kept fixed during the transition so that the value of the intensive variable can be calculated using the function of state.

<aside> 📌 SUMMARY: The work done when the value of an extensive variable changes can be calculated by integrating the conjugate intensive quantity.

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