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The important overall principle is that ''”Energy of all types changes from being localized to becoming dispersed or spread out, if it is not hindered from doing so. Entropy (or better, entropy change) is the quantitative measure of that kind of a spontaneous process: how much energy has been transferred/T or how widely it has become spread out at a specific temperature.''
=== Classical calculation of entropy ===
When entropy was first defined and used in 1865 the very existence of atoms was still controversial and there was no concept that temperature was due to the motional energy of molecules or that “heat” was actually the transferring of that motional molecular energy from one place to another. Entropy change, ΔS, was described in macro terms that could be measured, such as volume or temperature or pressure. The 1865 equation, which is still completely valid, is that ΔS = q (rev)/T. This can be expanded, part by part, in modern terms of how molecules are responsible for what is happening. Here is that equation expanded:
* ΔS = the entropy of a system (i.e., of a substance or a group of substances), after some motional energy (“heat”) has been transferred to it by fast moving molecules, minus the entropy of that system before any such energy was transferred to it. So, ΔS = S (final) – S (initial).
* Then, ΔS = S (final) – S (initial) = q, the motional energy (“heat”) that is transferred "reversibly" (rev) to the system from the surroundings (or from another system of in in contact with the first system) divided by T, the absolute temperature at which the transfer occurs = q (rev) / T.
** “Reversible” or “reversibly” (rev) simply means that T, the temperature of the system, has to stay (almost) exactly the same while any energy is being transferred to or from it. That’s easy in the case of phase changes, where the system absolutely must stay in the solid or liquid form until enough energy is given to it to break bonds between the molecules before it can change to a liquid or a gas. For example in the melting of ice at 273.0 K, no matter what temperature the surroundings are – from 273.1 K to 500 K or even higher, the temperature of the ice will stay at 273.0 K until the last molecules in the ice are changed to liquid water, i.e., until all the hydrogen bonds between the water molecules in ice are broken and new, less-exactly fixed hydrogen bonds between liquid water molecules are formed. This amount of energy necessary for ice melting per mole has been found to be 6008 joules at 273 K. Therefore, the entropy change per mole of q(rev)/T = 6008 J/273 K or 22 J/K.
** When the temperature isn't at the melting or boiling point of a substance no intermolecular bond-breaking is possible, and so any motional molecular energy (“heat”) from the surroundings transferred to a system raises its temperature, making its molecules move faster and faster. As the temperature is constantly rising, there is no longer a particular value of “T” at which energy is transferred. However, a "reversible" energy transfer can be measured at a very small temperature increase, and a cumulative total can be found by adding each of many many small temperature intervals or increments. For example, to find the entropy change (q(rev)/T) from 300 K to 310 K, measure the amount of energy transferred at dozens or hundreds of temperature increments, say from 300.00 K to 300.01 K and then 300.01 to 300.02 and so on, dividing the q by each T, and finally adding them all.
** Calculus can be used to make this calculation easier if the effect of energy input to the system is linearly dependent on the temperature change, as in simple heating of a system at moderate to relatively high temperatures. Thus, the energy being transferred “per incremental change in temperature” (the heat capacity, Cp), multiplied by the integral of dT/T from T(initial) to T(final), is directly given by ΔS = Cp ln T(final)/T(initial).
==Technical descriptions of entropy==
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