Ideal gas law
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The ideal gas law is the equation of state of an ideal gas. The state of an amount of gas is determined by its pressure, volume, and temperature. The equation has the form
- <math> pV = nRT \ </math>
where
- <math> p \ </math> is the pressure,
- <math> V \ </math> is the volume,
- <math> n \ </math> is the number of moles of gas,
- <math> R \ </math> is the gas constant, and
- <math> T \ </math> is the temperature.
The ideal gas law is most accurate for monoatomic gases and is favored at high temperatures and low pressures. It does not factor in the size of each gas molecule or the effects of intermolecular attraction. The more accurate Van der Waals equation takes these into consideration.
Proof
The ideal gas law can be proved using Boyle, Charles and Gay-Lussac laws.
Consider a volume <math>v_0</math> of gas. Let its state be defined as:
- <math>P_0 = 100 \ \mathrm{kPa} \,</math>
- <math>t_0 = 290 \ \mathrm{K}</math>
If this gas undergoes an isobaric process, its final volume will be:
- <math>v' = v_0(1 + \alpha t) \,</math>
and its temperature will be <math>t</math>.
If it then undergoes an isothermal process:
- <math>P_0v' = Pv \,</math>
So:
- <math> Pv = P_0v' \,</math>;
- <math> Pv = P_0v_0(1 + \alpha t) \,</math>;
- <math> Pv = {\frac{P_0 v_0}{290 \ \mathrm{K}}}T</math>;
where <math>{\frac{P_0 v_0}{290 \ \mathrm{K}}}</math> called <math>R</math>, is the universal gas constant. Using this notation we get:
- <math> pv = RT \,</math>
And multiplying both sides of the equation by n (numbers of moles):
- <math> Pnv = nRT \,</math>
Using the symbol <math>V</math> as a shorthand for <math>nv</math> (volume of n moles) we get:
- <math> pV = nRT \,</math>
See also
de:Thermische Zustandsgleichung idealer Gase es:Ley de los gases ideales fr:Calcul d'incertitude#la loi des gaz parfaits it:Equazione di stato dei gas perfetti nl:Algemene gaswet ja:理想気体の状態方程式 pl:Równanie Clapeyrona (stan gazu idealnego) sl:Splošna plinska enačba zh:理想气体状态方程