View source for Physics:Combined gas law

{{short description| Ideal gas law combining Charles's law, Boyle's law, and Gay-Lussac's law }}
{{Continuum mechanics}}
The '''combined gas law''' is a gas law that combines [[Physics:Charles's law|Charles's law]], [[Boyle's law]], and [[Physics:Gay-Lussac's law#Pressure-temperature law|Gay-Lussac's law]]. There is no official founder for this law; it is merely an amalgamation of the three previously discovered laws. These laws each relate one [[Physics:Conjugate variables (thermodynamics)|thermodynamic variable]] to another mathematically while holding everything else constant. Charles's law states that volume and temperature are directly proportional to each other as long as pressure is held constant. Boyle's law asserts that pressure and volume are inversely proportional to each other at fixed temperature. Finally, Gay-Lussac's law introduces a direct proportionality between temperature and pressure as long as it is at a constant volume. The inter-dependence of these variables is shown in the combined gas law, which clearly states that:

{{cquote|The ratio between the pressure-volume product and the temperature of a system remains constant.}}

This can be stated mathematically as:

:<math> \qquad \frac {PV}{T}= k </math>

where:

:{{math|''P''}} is the [[Physics:Pressure|pressure]],
:{{math|''V''}} is the [[Physics:Volume|volume]],
:{{math|''T''}} is the [[Physics:Thermodynamic temperature|temperature]] measured in [[Kelvin|kelvin]]s,
:{{math|''k''}} is a constant (with units of energy divided by temperature).

For comparing the same substance under two different sets of conditions, the law can be written as:

:<math> \qquad \frac {P_1V_1}{T_1}= \frac {P_2V_2}{T_2} </math>

The addition of [[Physics:Avogadro's law|Avogadro's law]] to the combined gas law yields the [[Physics:Ideal gas law|ideal gas law]].

==Derivation from the gas laws==


Boyle's Law states that the pressure-volume product is constant:
:<math>PV = k_1 \qquad (1)</math>
Charles's Law shows that the volume is proportional to the [[Physics:Absolute temperature|absolute temperature]]:
:<math>\frac{V}{T} = k_2\; \qquad (2)</math>
Gay-Lussac's Law says that the pressure is proportional to the absolute temperature:
:<math>P = k_3T \qquad (3)</math>

The algebraic manipulation cannot be done directly because Boyle did his experiments with constant [[Physics:Temperature|temperature]], Charles did them with constant [[Physics:Pressure|pressure]] and Lussac did them with constant [[Physics:Volume|volume]], therefore a more exact interpretation of the prior equations would be:

:<math>\begin{align}
PV &= k_1(T) \qquad (1')\\ 
\frac{V}{T} &= k_2(P) \qquad (2')\\ 
P/T &= k_3(V) \qquad (3'),\end{align}</math>
with <math>k_1,\;k_2,\;k_3</math> not being constants, but appropriate functions, mimicking the respective dependencies. 

Keeping this in mind, to carry the derivation on correctly, one must imagine the [[Physics:Gas|gas]] being altered by one process at a time. Say, starting to change only [[Physics:Pressure|pressure]] and [[Physics:Volume|volume]], according to Boyle's law, then:

:<math>P_1V_1=P_2V_2 \qquad (4).</math> After this process, the [[Physics:Gas|gas]] has parameters: <math>T_1</math>, <math>P_2</math> and <math>V_2.</math>

Then changing [[Physics:Volume|volume]] and [[Physics:Temperature|temperature]], according to Charles' law:

:<math>\frac{V_2}{T_1}=\frac{V_3}{T_2} \qquad (5).</math>  After this process, the state of the [[Physics:Gas|gas]] is: <math>T_2</math>, <math>V_3</math> and <math>P_2.</math>

Finally, [[Physics:Pressure|pressure]] and [[Physics:Temperature|temperature]] are changed obeying Gay-Lussac's Law:

:<math>\frac{P_2}{T_2}=\frac{P_3}{T_3} \qquad (6).</math>  After this process, the state is: <math>T_3</math>, <math>V_3</math> and <math>P_3.</math>

Applying [[Algebra|algebra]] yields:

:From equation (4) follows
:: <math>V_2=\frac{P_1V_1}{P_2} \qquad (7),</math> 
:substituting <math>V_2</math> into equation (5) yields
::<math>\frac{P_1V_1}{T_1P_2}=\frac{V_3}{T_2} \qquad (8),</math>
:and finally substituting <math>P_2</math> from (6) into (8) results in:
::<math>\frac{P_1V_1T_3}{T_1T_2P_3}=\frac{V_3}{T_2} \qquad (9),</math>
:which simplifies to:
::<math>\frac{P_1V_1}{T_1}=\frac{P_3V_3}{T_3} \qquad (10),</math>
or in other words
:<math>\frac{PV}{T}=k \qquad (11),</math>
where k is a [[Physics:Physical constant|physical constant]], and the dependencies  <math>k_1(T),\;k_2(P),\;k_3(V)</math> have vanished.

==Applications==
The combined gas law can be used to explain the mechanics where [[Physics:Pressure|pressure]], [[Physics:Temperature|temperature]], and [[Physics:Volume|volume]] are affected. For example: air conditioner<nowiki/>s, [[Physics:Refrigerator|refrigerator]]<nowiki/>s and the formation of [[Earth:Cloud|clouds]]. Also used in fluid mechanics and thermodynamics.

==See also==
*[[Physics:Dalton's law|Dalton's law]]
*[[Boyle's law|Boyle´s law]]
*[[Physics:Charles's law|Charles´s law]]
*[[Physics:Gay-Lussac's law|Gay-Lussac´s law]]

==Notes==
{{reflist}}

==Sources==
*Raff, Lionel. ''Principles of Physical Chemistry''. New Jersey: Prentice-Hall, 2001.

==External links==
* [http://chair.pa.msu.edu/applets/pvt/a.htm Interactive Java applet on the combined gas law] by Wolfgang Bauer
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[[Category:Gas laws]]

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