Introduction to entropy: Difference between revisions

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== Example of increasing entropy ==

{{Main article|Disgregation}}

Ice melting provides an example in which entropy increases in a small system, a thermodynamic system consisting of the surroundings (the warm room) and the entity of glass container, ice and water which has been allowed to reach [[thermodynamic equilibrium]] at the melting temperature of ice. In this system, some [[heat]] (''δQ'') from the warmer surroundings at {{Convert|298|K|C F|lk=on}} transfers to the cooler system of ice and water at its constant temperature (''T'') of {{Convert|273|K|C F}}, the melting temperature of ice. The entropy of the system, which is {{sfrac|δ''Q''|''T''}}, increases by {{sfrac|δ''Q''|273 K}}. The heat δ''Q'' for this process is the energy required to change water from the solid state to the liquid state, and is called the [[enthalpy of fusion]], i.e. Δ''H'' for ice fusion.

The entropy of the surrounding room decreases less than the entropy of the ice and water increases: the room temperature of {{val|298 |u=K}} is larger than {{val|273 |u=K}} and therefore the ratio, (entropy change), of {{sfrac|δ''Q''|298 K}} for the surroundings is smaller than the ratio (entropy change), of {{sfrac|δ''Q''|273 K}} for the ice and water system. This is always true in spontaneous events in a thermodynamic system and it shows the predictive importance of entropy: the final net entropy after such an event is always greater than was the initial entropy.

As the temperature of the cool water rises to that of the room and the room further cools imperceptibly, the sum of the {{sfrac|δ''Q''|''T''}} over the continuous range, "at many increments", in the initially cool to finally warm water can be found by calculus. The entire miniature 'universe', i.e. this thermodynamic system, has increased in entropy. Energy has spontaneously become more dispersed and spread out in that 'universe' than when the glass of ice and water was introduced and became a 'system' within it.

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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 6008 joules at {{val|273 |u=K}}. Therefore, the entropy change per mole is <math>\frac{q_\mathrm{rev}}{T} = \frac{6008\,\mathrm J}{273\,\mathrm K}</math>, or 22 J/K.

** 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>.