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.
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
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.
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 warm-water 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.
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