# Gas laws

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The early gas laws were developed at the end of the 18th century, when scientists began to realize that relationships between the pressure, volume and temperature of a sample of gas could be obtained which would hold for all gases. Gases behave in a similar way over a wide variety of conditions because to a good approximation they all have molecules which are widely spaced, and nowadays the equation of state for an ideal gas is derived from kinetic theory. The earlier gas laws are now considered as special cases of the ideal gas equation, with one or more of the variables held constant.

## Boyle's law

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Boyle's law shows that, at constant temperature, the product of the pressure and volume of a given mass of an ideal gas, assuming a closed system, is always constant. It was published in 1662. It can be determined experimentally using a pressure gauge and a variable volume container. It can also be derived from the kinetic theory of gases:if a container, with a fixed number of molecules inside, is reduced in volume, more molecules will hit a given area of the sides of the container per unit time, causing a greater pressure.

As a mathematical equation, Boyle's law is written as either:

${\displaystyle P\propto {\frac {1}{V}}}$
${\displaystyle PV=k_{1}}$
${\displaystyle P_{1}V_{1}=P_{2}V_{2}\,}$

where P is the pressure (Pa), V the volume (m3) of a gas, and k1 (measured in joules) is the constant from this equation—it is not the same as the constants from the other equations below.

## Charles' law

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Charles' Law, or the law of volumes, was found in 1787 by Jacques Charles. It says that, for a given mass of an ideal gas at constant pressure, the volume is directly proportional to its absolute temperature, asuming a closed system.

As a mathematical equation, Charles' law is written as either:

${\displaystyle V\propto T\,}$
${\displaystyle V/T=k_{2}}$
${\displaystyle V_{1}/T_{1}=V_{2}/T_{2}}$

where V is the volume (m3) of a gas, T is the temperature (measured in Kelvin) and k2 is the constant from this equation—it is not the same as the constants from the other equations below.

## Gay-Lussac's law

{{#invoke:main|main}} Gay-Lussac's law, or the pressure law, was found by Joseph Louis Gay-Lussac in 1809. It states that, for a given mass and constant volume of an ideal gas, the pressure exerted on the sides of its container is proportional to its temperature.

As a mathematical equation, Gay-Lussac's law is written as either:

${\displaystyle P\propto T\,}$
${\displaystyle P/T=k_{3}}$
${\displaystyle P_{1}/T_{1}=P_{2}/T_{2}}$

where P is the pressure (Pa), T is the temperature (measured in Kelvin), and k3 (is the constant from this equation—it is not the same as the constants from the other equations above.

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Avogadro's law states that the volume occupied by an ideal gas is proportional to the number of moles present in the container. This gives rise to the molar volume of a gas, which at STP is 22.4 dm3 (or litres). The relation is given by

${\displaystyle {\frac {V_{1}}{n_{1}}}={\frac {V_{2}}{n_{2}}}\,}$

where n is equal to the number of moles of gas (the number of molecules divided by Avogadro's Number).

## Combined and ideal gas laws

{{#invoke:main|main}} The combined gas law or general gas equation is formed by the combination of the three laws, and shows the relationship between the pressure, volume, and temperature for a fixed mass of gas:

${\displaystyle pV=k_{5}T\,}$

This can also be written as:

${\displaystyle \qquad {\frac {p_{1}V_{1}}{T_{1}}}={\frac {p_{2}V_{2}}{T_{2}}}}$

With the addition of Avogadro's law, the combined gas law develops into the ideal gas law:

${\displaystyle pV=nRT\,}$

where

p is pressure
V is volume
n is the number of moles
R is the universal gas constant
T is temperature (K)

where the constant, now named R, is the gas constant with a value of .08206 (atm∙L)/(mol∙K). An equivalent formulation of this law is:

${\displaystyle pV=kNT\,}$

where

p is the absolute pressure
V is the volume
N is the number of gas molecules
k is the Boltzmann constant (1.381×10−23 J·K−1 in SI units)
T is the temperature (K)

These equations are exact only for an ideal gas, which neglects various intermolecular effects (see real gas). However, the ideal gas law is a good approximation for most gases under moderate pressure and temperature.

This law has the following important consequences:

1. If temperature and pressure are kept constant, then the volume of the gas is directly proportional to the number of molecules of gas.
2. If the temperature and volume remain constant, then the pressure of the gas changes is directly proportional to the number of molecules of gas present.
3. If the number of gas molecules and the temperature remain constant, then the pressure is inversely proportional to the volume.
4. If the temperature changes and the number of gas molecules are kept constant, then either pressure or volume (or both) will change in direct proportion to the temperature.

## Other gas laws

• Graham's law states that the rate at which gas molecules diffuse is inversely proportional to the square root of its density. Combined with Avogadro's law (i.e. since equal volumes have equal number of molecules) this is the same as being inversely proportional to the root of the molecular weight.
${\displaystyle P_{total}=P_{1}+P_{2}+P_{3}+...+P_{n}\equiv \sum _{i=1}^{n}P_{i}\,}$,

OR

${\displaystyle P_{\mathrm {total} }=P_{\mathrm {gas} }+P_{\mathrm {H_{2}O} }\,}$

where PTotal is the total pressure of the atmosphere, PGas is the pressure of the gas mixture in the atmosphere, and PH2O is the water pressure at that temperature.

At constant temperature, the amount of a given gas dissolved in a given type and volume of liquid is directly proportional to the partial pressure of that gas in equilibrium with that liquid.
${\displaystyle p=k_{\rm {H}}\,c}$

## References

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