52 Kinetic Theory of Gases and Gas Laws
If the data is recorded for a gas A we get line-1 shown in Here, P V and T stand for pressure, volume and temperature
figure-2.4 and if we repeat the experiment with another gas B, respectively and R is a constant that is same for all gases and
with the same initial volume and temperature, we get the same so is called the universal gas constant and n are the number of
results and obtain line-1. But if we use gas B with different moles of gas. If pressure is measured in SI units of units of
initial volume, we obtain data producing a new line, line-2. pascal, volume in cubic meters, and temperature in kelvins, then
Whatever gas we use the behaviour is same. The graph will R has a value 8.314 Joule/mole-K. If in a state gas pressure,
always be a straight line. volume and temperature are P1, V1 and T1 and after some
experiment on same sample of gas if its pressure, volume and
Here we can see that these straight lines if extended intersect temperature are P2, V2 and T2, then from equation-(2.7) we have
with temperature axis at – 273.15ºC. Theoretically it shows that P1V1 P2V2
the volume of gas become zero at this temperature. The reason = … (2.8)
T1 T2
already we’ve assumed that the size of molecules is negligible
As number of moles in initial and final state are equal.
and at – 273.15ºC or 0 K temperature, the kinetic energy of
molecules become zero or all motions are frozen at 0 K
The amount of a gas is generally measured in moles, given as n.
temperature thus no movement is there in gas molecules of
A mole (mol) is the amount of material whose mass in grams is
negligible size at this temperature. If the graph shown in figure-
numerically equal to the molecular mass of substance. For
2.4 is again plotted with Kelvin scale we get one as shown in
example molecular mass of O2 is 32, Thus a mole of oxygen is 32
figure-2.5.
grams.
This scale is absolute scale with the sense that its zero
2.2.4 Avogadro’s Number and Avogadro Hypothesis
(– 273.15ºC) is the lower limit for temperature and in practical
nature to attain a temperature below this is not possible due to
We’ve discussed that the gas constant R, has the same value
the reason discussed above. In further analysis of gases we
for all gases. This fact was first recognised in a slightly different
will use Kelvin scale.
form, by an Italian Scientist Ameodeo Avogadro. Avogadro
V stated that equal volumes of gas at the same pressure and
B
A, temperature contain equal numbers of molecules. This statement
1
e- is called Avogadro’s hypothesis. We can see that this statement
lin B
e-2 is consistent with R being the same for all gases.
Volume
lin
The number of molecule in a mole is known as Avogadro’s
number, NA. Although Avogadro was not able to actually
O
T determine the value of NA, several methods have been devised
Temperature (K)
to measure NA and the acceptable value found is
Figure 2.5
NA = 6.023 × 1023 molecules/mole
Thus if temperature is expressed in Kelvins, we find that when
the pressure is held constant, the volume is proportional to the Thus in n moles of a gas total number of molecules of the gas
temperature. This statement in the law of Charles and Gay- are
Lussac. This can be expressed mathematically as N = n NA … (2.9)
V1 T1
= … (2.5) or number of moles of a gas can be given as
V2 T2
N m N m
V n= = = … (2.10)
or = constant … (2.6) N A m N A M
T
Here m is the mass of each molecule, m is the total mass of gas
As with Boyle’s law, the amount of gas also must be held and M is the mass of 1 mole of molecules of gas i.e. its molecular
constant for equation-(2.5) and (2.6) to be valid. mass.
2.2.3 Ideal Gas Law From gas law, we have
Boyle’s law and the law of Charles and Gay-Lussac are the PV = n R T
Particular cases of a more general expression called the “ideal N m
gas law”. It can be written as or PV = RT = RT
NA M
PV = n R T … (2.7) or PV = N k T … (2.11)
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