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 [Pa],
- <math>\ V </math> is the volume [m3],
- <math>\ n </math> is the number of moles of gas [mol],
- <math>\ R </math> is the gas constant 8.314472 [m3·Pa·K-1·mol-1], and
- <math>\ T </math> is the temperature in kelvin [K].
The gas constant ("R") is dependent on what units are used in the formula. The value given above, 8.314472, is for the SI units of Pascal-Meters3 per per mole-Kelvin. Another value for R is .0821 L atm mol^-1 K^-1)
The ideal gas law is most accurate for monatomic 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.
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[edit] Alternate Forms
Considering that the number of moles (<math> n </math>) could also be given in mass, sometimes you may wish to use an alternate form of the ideal gas law. This is particularly useful when asked for the ideal gas law approximation of a known gas. Consider that the number of moles (<math> n </math>) is equal to the mass (<math> m </math>) divided by the molar mass (<math> M </math>), such that:
- <math> n = {\frac{m}{M}} </math>
Then, replacing <math> n </math> gives:
- <math>\ pV = {\frac{mRT}{M}} </math>
In thermodynamics and physics, when something is referred to as specific, it simply means the value per unit mass. In the case of the gas constant, the specific gas constant (<math> r </math>) would be <math> R </math> divided by the molar mass of the gas in question:
- <math> r = {\frac{R}{M}} </math> or <math> R = rM </math> (where <math> r </math> is the specific gas constant)
Replacing with <math> r </math> in the above formula yields:
- <math>\ pV = mrT </math> (the molar masses divide out) or <math>\ pV = nrMT </math>
Since density (ρ) equals mass over volume, it is possible to substitute volume for grams over density (V = g/ρ), and make appropriate changes.
[edit] Proof
[edit] Empirical
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>
Firstly, 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>.
Secondly, 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>
[edit] Theoretical
The ideal gas law can also be derived from first principles using the kinetic theory of gases, if the molecules are assumed to be hard spheres.
[edit] See also
da:Idealgasligning de:Thermische Zustandsgleichung idealer Gase es:Ley de los gases ideales fr:Calcul d'incertitude#la loi des gaz parfaits ko:이상 기체 상태방정식 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) ru:Уравнение состояния идеального газа sl:Splošna plinska enačba sv:Ideala gaslagen uk:Рівняння стану ідеального газу zh:理想气体状态方程
