Derivation of the
Ideal Gas Law
We have now identified the models of
how we look at atoms and molecules,
studied reactions they undergo and have
learned how energy is involved in these
processes. Now we will turn our study to
the different phases in which matter exists.
You will find that these units on
phases of matter and the kinetic-molecular
theory will be quite extensively tested in the
multiple-choice section of the national AP
exam; especially the unit on gases. Be
forewarned.
Get out AP equation sheet
Characteristics of Gases
1. Non metallic compounds
2. Simple formulas w/ low M.W.
3. A.k.a. Vapors for substances that are
normally (s) or (l) at room temp.
4. Expand to fill their container
5. Highly compressible
6. Form homogeneous mixtures
7. Relatively far apart (behave
independent of other molecules)
•Volume (V)
= The total space of a container that
gases occupy due to the free random
motion of the gas molecules
(reported in liters, L)
•Number of moles (n)
= The total number of gas
molecules in a collection of
particles. (reported in moles, mol)
Factors that Influence
Gas Behavior
•Temperature (T)
= The indirect measure of the
average kinetic energy of a
collection of particles (on the
Kelvin scale, K)
•Pressure (P)
= The measure of the number of
collisions between gas particles
and a unit area of the walls of its
container (reported in KPa or
atm)
P=F/A
having the units (N/m2)
1N/m2 = 1 Pascal (Pa)
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Pressure of air is measured
with a BAROMETER
(developed by Torricelli in 1643)
Hg rises in tube until force
of Hg (down) balances the
force of atmosphere
(pushing up).
Aneroid Barometer
Pressure
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Column height measures
pressure of the
atmosphere
• 1 standard atm
= 760 mm Hg
= 760 torr
= 29.92 inches Hg
= 14.70 psi
= about 34 feet of water
• SI unit is PASCAL, Pa,
– where 1 atm = 101.325 kPa
In order to measure the pressure
exerted by an enclosed gas, scientist
use a device called an manometer.
Two types of manometers:
1. Open-end manometer- used for gas
pressures near 1 atm.
2. Closed-end manometer- used for
gas pressures greatly over or under
1 atm.
1.The pressure of a gas is
measured at 49.0 torr in an
open-end manometer. Describe
this pressure in the units of
atmospheres and describe the
mercury level in the manometer
in respect to the pressure of the
gas and the atmosphere.
The Classical Gas Laws
•Boyle’s Law
The volume of a
fixed quantity of gas at
constant temperature is
inversely proportional
to the pressure
Robert Boyle
(1627-1691). Son of
Earl of Cork,
Ireland.
2
(L, atm)
•Boyle’s Law
The volume of a
fixed quantity of gas at
constant temperature is
inversely proportional to
the pressure
V = 1/P
VP = k
V1P1 = V2P2
y=mx+b
1
b
P
k
or V 
P
V m
2. Baby George was given a 4.53 dm3
helium filled metallic balloon on a
bright, sunny day when the
barometric pressure was 768 torr.
That night a storm front moved in
and the pressure dropped to 732 torr.
What is the expected volume of the
balloon at these new barometric
conditions?
(L, K)
•Charles’ Law
V = k’ T
The volume of
a fixed amount of
gas maintained at
constant pressure is
proportional to its
absolute
temperature.
V1/ T1 = V2/ T2
Jacques Charles (17461823). Isolated boron
and studied gases.
Balloonist.
He
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V(L)
4
Y=mx+b
V=mT+b
V = k’ T
CH4
3
H2O
2
H2
1
N2O
-200
0
T(0C)
•Charles’ Law
The volume of a fixed amount of gas maintained at
constant pressure is proportional to its absolute temperature.
3. Baby George received a larger 6.19
dm3 helium filled metallic balloon
on top of Pike's Peak at noon when
the temperature was 200C. That
night, the temperature dropped to
-100C. What volume did the balloon
occupy at that temperature?
200
3
Avogadro’s Law
• The volume of a gas at constant temperature
and pressure is directly proportional to the
number of moles of the gas.
• Mathematically, this means
V = kn
V1/ n1 = V2/ n2
Y=mx+b
V=mn+b
V = k’’ n
Avogadro’s
Hypothesis
At constant temperature and
pressure, equal volumes of gases
contain equal number of particles
It was found that one mole of any
gas occupies a volume equal to
22.414 L at STP, or 0oC and 1 atm
(Avagadro’s law)
Ideal-Gas Equation
• So far we’ve seen
that V  1/P (Boyle’s law)
V
V  T (Charles’s law)
V  n (Avogadro’s law)
• Combining these, we
get
nT
V
P
n
Ideal-Gas Equation
The relationship
then becomes
or
V
V=R
nT
P
nT
P
PV = nRT
With the addition of a proportionality constant
(R).
“R” is the ideal gas constant
describing the volume of one mole of a gas
at 1 atm and 0oC
L  atm
R  0.0821
mol  K
L  kPa
 8.314
mol  K
L  torr
 62.36
mol  K
= 8.314
J
mol • K
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4. A sample of hydrogen gas has a
volume of 8.56 L at a
temperature of 0oC and a
pressure of 1.5 atm. Calculate
the mass of hydrogen present in
the sample.
Relevance of the Ideal Gas Law
The Ideal gas law provides a
constant set of conditions to which we
can compare any gas sample.
However, if any one variable is
held constant, the combined gas law
can be utilized, where:
P1V1/ n1T1 = P2V2/ n2T2
5. A sample of diborane gas (B2H6), a
substance that burst into flame when
exposed to air, has a pressure of 345
torr at a temperature of –15.0oC and
a volume of 3.48 L. If the conditions
changed the temperature to 36.0oC
and the pressure to 468 torr, what
will be the volume?
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