A state function of a system in equilibrium is a macroscopic value which is a pure function of the state of the system. So, for example, we know that any two samples of the same quantity of a specific gas at given temperature and pressure will occupy the same volume. In some cases ( e.g. an ideal gas where PV=RTPV=RT ) we may know the function precisely. But even in cases where we don’t know the exact form of the function, we can still be sure it exists and use this in our arguments.
In order to define the state of a system precisely, we simply need the values of a sufficient number of state functions, from which we can in principle derive the rest. Some are easier to measure or control than others. In practice we commonly we use two of temperature, pressure, and volume, but in theoretical terms, it is often most useful to use Entropy SS and Volume VV.
Other common state functions are Internal Energy UU, Enthalpy H=U+PVH=U+PV, Free Energy F=U−TSF=U−TS , and Gibbs Free Energy G=U−TS+PVG=U−TS+PV.
It is often useful to produce a 2d graph with our two chosen state functions as the axes. Every point represents a specific state of the substance. Lines connecting equal values of other state functions can be drawn ( like contours on a map ).
Transformations between states can be shown as paths on the state diagram.
What a state diagram does not tell us is the net effect on the rest of the world of a state change. This is because state changes can be reversible or irreversible depending on whether the change is accompanied by a total change in the entropy of (system + rest of world) which is 0 or greater.
In any event, the fundamental equation is dU=TdS−PdVdU=TdS−PdV which applies to any situation, reversible or not. There is a similar, but different equation which also applies to any situation, i.e. dU=q+wdU=q+w where qq represents heat and ww represents work. qq and ww are not differentials as they are not path independent. In the reversible case, q=TdSq=TdS and w=−PdVw=−PdV, whereas in the non-reversible case w=−PdV+Xw=−PdV+X and q=TdS−Xq=TdS−X where XX represents the extra work done on the system in an irreversible situation, i.e. the work turned into heat ‘unnecessarily’. XX thus has the effect of increasing the work done to the system ( or reducing the work taken from the system ) in making the state change.
The second law of thermodynamics can therefore be seen as requiring X>=0X>=0 under all circumstances. When X=0X=0 the situation is reversible, and entropy may be transferred to or from the system but does not increase in the universe as a whole.
Some example processes include:
- A reversible isothermal expansion in which our system is in contact with a heat bath of infinitesimally higher temperature. This causes heat to flow to our system from the bath, and the energy will allow the system to expand into the environment at its prevailing pressure, doing work in the process. As the sample expands, the pressure will fall, limiting the energy which can be transferred. Note that to be reversible, the external pressure must be infinitesimally less than the pressure of the substance at all times. For a perfect gas, internal energy does not vary with volume so dU=0=q+w=TdS−PdVdU=0=q+w=TdS−PdV Since T is constant, the change in entropy of the system is proportional to the change in volume.
- A work-free isothermal expansion, e,g, where the system expands into a larger vacuum chamber. In this case for a perfect gas dU=0=wdU=0=w and so qq is also 0 and no heat is taken from the environment. So the entropy of the sample goes up, that of the environment remains unchanged, but TdSTdS is the same as in the reversible case, as is PdVPdV
For an adiabatic reversible process, the substance represented by the state diagram is assumed totally isolated, and thus for such processes, the entropy remains constant. However, for an isothermal process in which the substance can exchange heat with a reservoir, the entropy can fall, as there will then be a corresponding rise in the entropy of the reservoir.
Although it is the ‘basic’ equation, in the sense that it is the one easiest to derive from physical principles, the above equation for internal energy in terms of volume and entropy is very inconvenient in most practical situations. In particular, a major use of thermodynamics is in chemistry, where it is pressure, not volume which is usually considered a control parameter, and temperature not entropy which complements it. It is entirely mathematical manipulation with no additional physical input which allows us to cast the equation above in terms of enthalpy, hemholtz free energy and gibbs energy.
