Thermodynamics and Propulsion  

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The situation just described is a special case of an important principle concerning entropy changes, irreversibility and the loss of capability to do work. We thus now develop it in a more general fashion, considering an arbitrary system undergoing an irreversible state change, which transfers heat to the surroundings (for example the atmosphere), which can be assumed to be at constant temperature, . The change in internal energy of the system during the state change is . The change in entropy of the surroundings is (with the heat transfer to the system)
Now consider restoring the system to the initial state by a reversible process. To do this we need to do work, , on the system and extract from the system a quantity of heat, . (We did this, for example, in ``undoing'' the free expansion process.) The change in internal energy is (with the quantities and both regarded, in this example, as positive for work done by the surroundings and heat given to the surroundings)^{6.2}.
In this reversible process, the entropy of the surroundings is changed by
For the combined changes (the irreversible state change and the reversible state change back to the initial state), the energy change is zero because the energy is a function of state,
Thus,
For the system, the overall entropy change for the combined process is zero, because the entropy is a function of state,
The total entropy change is thus only reflected in the entropy change of the surroundings:
The surroundings can be considered a constant temperature heat reservoir and their entropy change is given by
We also know that the total entropy change, for system plus surroundings is,
The total entropy change is associated only with the irreversible process and is related to the work in the two processes by
The quantity represents the extra work required to restore the system to the original state. If the process were reversible, we would not have needed any extra work to do this. It represents a quantity of work that is now unavailable because of the irreversibility. The quantity can also be interpreted as the work that the system would have done if the original process were reversible. From either of these perspectives we can identify as the quantity we denoted previously as , representing lost work. The lost work in any irreversible process can therefore be related to the total entropy change (system plus surroundings) and the temperature of the surroundings by
To summarize the results of the above arguments for processes where heat can be exchanged with the surroundings at :
Muddy Points
Is path dependent? (MP 6.11)
Are and the and going from the final state back to the initial state? (MP 6.12)
Douglas Quattrochi 20060806