Changes

:<math>PV = constant</math>

:<math>PV = constant</math>

where the constant depends on the amount and type of the gas sample

where the constant depends on the amount and type of the gas sample.

 

[[Charles' Law]], formulated in the 1780s, stated that, for a given sample of gas at a fixed pressure, the volume was directly proportional the "absolute" temperature.  That is,

[[Charles' Law]], formulated in the 1780s, stated that, for a given sample of gas at a fixed pressure, the volume was directly proportional the "absolute" temperature.  That is,

:<math>V/T = constant</math>

:<math>V/T = constant</math>

There are a number of ways of stating this.  A mole is [[Avogadro's number]] (6.022x10²³) of molecules.  The gas law can be restated in terms of the number of molecules:

There are a number of ways of stating this.  A mole is [[Avogadro's number]] (6.022x10²³) of molecules.  The gas law can be restated in terms of the number of molecules:

:<math>PV = nkT\,</math>

:<math>PV = nkT\,</math>

where <math>n\,</math> is the number of molecules and <math>k\,</math> is ''Boltzmann's constant'' (1.38x10^-23 joules per kelvin).

where <math>n\,</math> is the number of molecules and <math>k\,</math> is ''Boltzmann's constant'' (1.38x10^-23 joules per kelvin).  It is just the universal gas constant <math>R\,</math> scaled by Avogadro's number.  Since it makes no reference to artificial units like grams, physicists consider it to be more theoretically significant than the gas constant, and it shows up in many physical formulas (quantum mechanics and statistical mechanics, for example) that are not related to gas behavior.

Scientists were now fairly close to figuring out thermodynamics.  They just needed the kinetic theory and statistical mechanics.  They still didn't know why heat only flows "downhill"&mdash;it was still just an observed fact.  And they didn't know why "heat engines", that is, things that turn heat (e.g. steam) into mechanical energy, aren't very efficient.

Scientists were now fairly close to figuring out thermodynamics.  They just needed the kinetic theory and statistical mechanics.  They still didn't know why heat only flows "downhill"&mdash;it was still just an observed fact.  And they didn't know why "heat engines", that is, things that turn heat (e.g. steam) into mechanical energy, aren't very efficient.

==[[Second Law of Thermodynamics]]==

==[[Second Law of Thermodynamics]]==

James Watt's steam engine drew heat from a specific source and converted some of it to useful work; the remainder of the heat was transferred to a cooler reservoir.  In 1824 a French engineer, N.L. Sadi Carnot, proposed the Carnot cycle<ref>[https://en.wikiversity.org/wiki/Carnot_engine] Carnot Engine</ref>, consisting of two isothermal processes (involving a constant temperature) and two adiabatic processes (no heat is gained or lost).  The result was, in theory, the most efficient-working heat engine cycle of any kind, one involving processes that must be reversible and involve no change in entropy.  What was discovered in practice was the second law of thermodynamics, which states (in one of its various formulations) that [[entropy]] in an isolated system cannot decrease, and that ''irreversible processes'' can only make it increase.<ref>Gregory H. Wannier, ''Statistical Physics'', John Wiley & Sons, New York, 1966</ref> An equivalent formulation states that heat cannot spontaneously flow from a cooler body to a hotter body.  Clausius stated in what is known as the Clausius statement that <ref>S. J. Blundell & K. M. Blundell, ''Concepts in Thermal Physics'', Oxford University Press, New York, 2016</ref>

James Watt's steam engine drew heat from a specific source and converted some of it to useful work; the remainder of the heat was transferred to a cooler reservoir.  In 1824 a French engineer, N.L. Sadi Carnot, proposed the Carnot cycle<ref>[https://en.wikiversity.org/wiki/Carnot_engine Carnot Engine]</ref>, consisting of two isothermal processes (involving a constant temperature) and two adiabatic processes (no heat is gained or lost).  The result was, in theory, the most efficient-working heat engine cycle of any kind, one involving processes that must be reversible and involve no change in entropy.  What was discovered in practice was the second law of thermodynamics, which states (in one of its various formulations) that [[entropy]] in an isolated system cannot decrease, and that ''irreversible processes'' can only make it increase.<ref>Gregory H. Wannier, ''Statistical Physics'', John Wiley & Sons, New York, 1966</ref> An equivalent formulation states that heat cannot spontaneously flow from a cooler body to a hotter body.  Clausius stated in what is known as the Clausius statement that <ref>S. J. Blundell & K. M. Blundell, ''Concepts in Thermal Physics'', Oxford University Press, New York, 2016</ref>

{{cquote|No process is possible whose sole result is the transfer of heat from a colder to a hotter body.}}

{{cquote|No process is possible whose sole result is the transfer of heat from a colder to a hotter body.}}

{{cquote|No process is possible whose sole result is the complete conversion of heat into work}}

{{cquote|No process is possible whose sole result is the complete conversion of heat into work}}

What it meant was heat transfers to cooler temperatures, and not the other way around, which led to the concept of entropy.  An idealised formula for an increase in entropy when heat is applied reads:

What it meant was heat transfers to cooler temperatures, and not the other way around.  But this was still just a carefully worked out summary of experimental observation.  While such summaries of observation are important for the progress of science, people still didn't know '''why''' this was true.  Nevertheless, the development of Carnot's theory used that observation to explain why heat engines have limited efficiency.

<center><big><math> \delta S_{rev} = Q/T </math></big></center>

 

Where <math>\delta S_{rev}</math> is the small change in entropy caused by a reversible process, <math>Q</math> is the added heat, and <math>T</math> is the absolute temperature.  This increase in entropy must be at least <math>q/T</math>, but the entropy change itself <math>\delta S</math> is always greater.

Making progress on the theory required the development of the concept of entropy, and the development of statistical mechanics.  To see what entropy is about, consider its thermodynamical definition as a differential.  Entropy is generally symbolized with a capital S, and heat energy with a capital Q.  While defining entropy in terms of its derivative rather than an actual absolute definition may seem to leave something to be desired (it leaves a "constant of integration" unspecified), that generally doesn't matter.  This standard definition is used:

::<math>dS = dQ/T</math>

The change in entropy of some object is the change in heat energy divided by the temperature at which the change takes place.  The unit of thermodynamic entropy is the Joule per Kelvin.  This is an "extensive" measure.  To get the entropy for a given substance, independently of the size of the sample, it has to be divided by the size of the sample.  So the unit of entropy for a substance (e.g. ice) is Joules per Kelvin per mole.  Or per gram, or per atom, or per liter, or whatever.

 

::<math>dQ_{coffee} = -50\qquad\qquad\qquad\qquad dQ_{room} = 50</math>

::<math>dQ_{coffee} + dQ_{room} = 0</math>

(These would actually be instantaneous time derivatives.)  Energy is conserved.  But now consider the equation for entropy.  It the coffee is at 310 Kelvins and the room at 290 Kelvins:

::<math>dS_{coffee} = -50/310 = -0.1613\qquad\qquad\qquad\qquad dS_{room} = 50/290 = 0.1724</math>

:::Entropy always increases, or stays the same.  It does not decrease.

This is still just a nicely formulated statement of observations.  To see why this is the correct definition of entropy, and why this law is correct, statistical mechanics must be developed.  This was done by Maxwell, Boltzmann, Clausius, and others during the 19^th century.

 

For a continuation of this, see the [[Second Law of Thermodynamics]] article.

==[[The Third Law of Thermodynamics]]==

==[[The Third Law of Thermodynamics]]==

Also known as ''Nernst's Law'', states that it is not possible to bring any system to the [[absolute zero]] of temperature in a finite number of operations. Also stated as follows: The entropy of a perfect crystal at absolute zero is zero.

Also known as ''Nernst's Law'', states that it is not possible to bring any system to the [[absolute zero]] of temperature in a finite number of Carnot cycles. Also stated as follows: The entropy of a perfect crystal at absolute zero is zero.

These laws tell us to what constraints ''any'' system is subject. For example, it allows us to calculate the maximum possible efficiency of an [[engine]] once we know the temperature at which it operates.

These laws tell us to what constraints ''any'' system is subject. For example, it allows us to calculate the maximum possible efficiency of an [[engine]] once we know the temperature at which it operates.