Gas laws
This article outlines the historical development of the laws
describing ideal gases. For a detailed description of the ideal gas
laws and their further development, see Ideal gas, Ideal gas law
and Gas.
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
Boyle's law shows that, at constant temperature, the product of
an ideal gas's pressure and volume 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
found through the use of logic; if a container, with a fixed number
of molecules inside, is reduced in volume, more molecules will hit
the sides of the container per unit time, causing a greater
pressure.
As a mathematical equation, Boyle's law is:
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.
This is known as Boyle's law which states: the volume of a given
mass of gas is inversely proportional to its pressure, if the
temperature remains constant. Mathematically this is:
V = k/P
where k is a constant (NOT Boltzmann's constant or Coulomb’s
constant).
Charles' law
Charles's Law, or the law of volumes, was found in 1787 by
Jacques Charles. It says that, for an ideal gas at constant pressure,
the volume is directly proportional to its temperature.
\frac{V_1}{T_1}=\frac{V_2}{T_2} \,
Gay-Lussac's law
Gay-Lussac's law, or the pressure law, was found by Joseph Louis
Gay-Lussac in 1809. It states that the pressure exerted on the
sides of a container by an ideal gas of fixed volume is proportional
to its temperature.
\frac{P_1}{T_1}=\frac{P_2}{T_2}
Avogadro's law
Avogadro's law states that the volume occupied by an ideal gas is
proportional to the number of moles (or molecules) 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
\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
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:
PV = k_5T \,
This can also be written as:
\qquad \frac {p_1V_1}{T_1}= \frac {p_2V_2}{T_2}
With the addition of Avogadro's law, the combined gas law
develops into the ideal gas law:
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:
1. 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:
If temperature and pressure are kept constant, then the volume
of the gas is directly proportional to the number of molecules of
gas.
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.
If the number of gas molecules and the temperature remain
constant, then the pressure is inversely proportional to the
volume.
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.
Dalton's law of partial pressures states that the pressure of a
mixture of gases simply is the sum of the partial pressures of the
individual components. Dalton's Law is as follows:
P_{total} = P_1 + P_2 + P_3 + ... + P_n \equiv \sum_{i=1}^n P_i \,,
OR
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.
Henry's law states that:
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.
p = k_{\rm H}\, c
Boyle's Law
Torricelli's experiment did more than just show that air has
weight; it also provided a way of creating a vacuum because the
space above the column of mercury at the top of a barometer is
almost completely empty. (It is free of air or other gases except a
negligible amount of mercury vapor.) Torricelli's work with a
vacuum soon caught the eye of the British scientist Robert Boyle.
Boyle's most famous experiments with gases dealt with what he
called the "spring of air." These experiments were based on the
observation that gases are elastic. (They return to their original
size and shape after being stretched or squeezed.) Boyle studied
the elasticity of gases in a J-tube similar to the apparatus shown in
the figure below. By adding mercury to the open end of the tube,
he trapped a small volume of air in the sealed end.
Boyle studied what happened to the volume of the gas in the
sealed end of the tube as he added mercury to the open end.
Boyle noticed that the product of the pressure times the volume
for any measurement in this table was equal to the product of the
pressure times the volume for any other measurement, within
experimental error.
P1V1 = P2V2
This expression, or its equivalent,
equation
is now known as Boyle's Law.
Amonton's Law
Toward the end of the 1600s, the French physicist Guillaume
Amontons built a thermometer based on the fact that the pressure
of a gas is directly proportional to its temperature. The relationship
between the pressure and the temperature of a gas is therefore
known as Amontons' law.
P
T
Amontons' law explains why car manufacturers recommend
adjusting the pressure of your tires before you start on a trip. The
flexing of the tire as you drive inevitably raises the temperature of
the air in the tire. When this happens, the pressure of the gas inside
the tires increases.
Amontons' law can be demonstrated with the apparatus shown in
the figure below, which consists of a pressure gauge connected to a
metal sphere of constant volume, which is immersed in solutions
that have different temperatures.
The apparatus for demonstrating Amonton's law consists
of .
The following data were obtained with this apparatus.
In 1779 Joseph Lambert proposed a definition for absolute zero on
the temperature scale that was based on the straight-line
relationship between the temperature and pressure of a gas shown
in the figure above.
He defined absolute zero as the temperature at which the pressure
of a gas becomes zero when a plot of pressure versus temperature
for a gas is extrapolated. The pressure of a gas approaches zero
when the temperature is about -270�C. When more accurate
measurements are made, the pressure of a gas extrapolates to zero
when the temperature is -273.15�C. Absolute zero on the Celsius
scale is therefore -273.15�C.
The relationship between temperature and pressure can be greatly
simplified by converting the temperatures from the Celsius to the
Kelvin scale.
TK = ToC + 273.15
When this is done, a plot of the temperature versus the pressure of
a gas gives a straight line that passes through the origin. Any two
points along the line therefore fit the following equation.
It is important to remember that this equation is only valid if the
temperatures are converted from the Celsius to the Kelvin scale
before calculations are done.
Charles' Law
On 5 June 1783, Joseph and Etienne Montgolfier used a fire to
inflate a spherical balloon about 30 feet in diameter that traveled
about a mile and one-half before it came back to earth. News of
this remarkable achievement spread throughout France, and
Jacques-Alexandre-Cesar Charles immediately tried to duplicate
this performance. As a result of his work with balloons, Charles
noticed that the volume of a gas is directly proportional to its
temperature.
V T
This relationship between the temperature and volume of a gas,
which became known as Charles' law, provides an explanation of
how hot-air balloons work. Ever since the third century B.C., it has
been known that an object floats when it weighs less than the fluid
it displaces. If a gas expands when heated, then a given weight of
hot air occupies a larger volume than the same weight of cold air.
Hot air is therefore less dense than cold air. Once the air in a
balloon gets hot enough, the net weight of the balloon plus this hot
air is less than the weight of an equivalent volume of cold air, and
the balloon starts to rise. When the gas in the balloon is allowed to
cool, the balloon returns to the ground.
Charles' law can be demonstrated with the apparatus shown in the
figure below. A 30-mL syringe and a thermometer are inserted
through a rubber stopper into a flask that has been cooled to 0�C.
The ice bath is then removed and the flask is immersed in a warmwater bath. The gas in the flask expands as it warms, slowly
pushing the piston out of the syringe. The total volume of the gas
in the system is equal to the volume of the flask plus the volume of
the syringe.