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* <math>\Delta S</math> is the change in entropy of a system (some physical substance of interest) after some motional energy ("heat") has been transferred to it by fast-moving molecules. So, <math>\Delta S = S_\mathrm{final} - S _\mathrm{initial}</math>.
* Then, <math> \Delta S = S_\mathrm{final} - S _\mathrm{initial} = \frac{q_\mathrm{rev}}{T}</math>, the quotient of the motional energy ("heat") q that is transferred "reversibly" (rev) to the system from the surroundings (or from another system in contact with the first system) divided by T, the absolute temperature at which the transfer occurs.
** "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 is 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 {{val|273.15|ul=K}}, no matter what temperature the surroundings are – from {{val|273.20|u=K}} to {{val|500|u=K}} or even higher, the temperature of the ice will stay at {{val|273.15|u=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 {{val|6008
** When the temperature is not 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 small temperature intervals or increments. For example, to find the entropy change <math>\frac{q_\mathrm{rev}}{T}</math> from {{val|300|u=K}} to {{val|310|u=K}}, measure the amount of energy transferred at dozens or hundreds of temperature increments, say from {{val|300.00|u=K}} to {{val|300.01|u=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, <math>C_p</math>), multiplied by the [[integral]] of <math>\frac{dT}{T}</math> from <math>T_\mathrm{initial}</math> to <math>T_\mathrm{final}</math>, is directly given by <math>\Delta S = C_p \ln\frac{T_\mathrm{final}}{T_\mathrm{initial}}</math>.
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