Ideal Gases
3.3.1 State the macroscopic gas laws relating pressure, volume and temperature.
Boyle’s Law – For a gas at a constant temperature the volume and the pressure are
inversely proportional:
(1)
P ∝ 1/V
Plotting pressure vs. volume produces the
Plotting pressure vs. inverse volume
following graph:
produces a straight line graph:
If a thermodynamic system changes, but its temperature remains constant then the
following the initial pressure and volume are related to the final pressure and volume
by the following equation:
(2)
P1V1=P2V2
Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the
gas is directly proportional to its absolute temperature.
(3)
V∝T
If a thermodynamic system changes, but its mass and pressure are held constant the
initial volume and temperature are related to the final volume and temperature by the
following equation:
(4)
V1/T1=V2/T2
Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute
pressure of the gas is directly proportional to its absolute temperature.
(5)
P∝T
If a thermodynamic system changes, but the volume are held constant the initial
pressure and temperature are related to the final pressure and temperature by the
following equation:
(6)
P1T1=P2T2
All of the above mentioned laws required special circumstances that are not often
satisfied. A more general relationship between the volume, pressure and temperature
can be found:
(7)
P1V1/T1=P2V2/T2
This still does not take into account a change in mass. If the temperature and pressure
of a gas is held constant and more gas is added (mass increases) the volume must
increase. If the volume and temperature of a gas are held constant while more gas is
added the pressure must increase. If we combine these experimental observations we
can write an even more general relation:
(8)
P1V1 / m1T1=P2V2 / m2T2
3.3.2 Define the terms mole and molar mass
In general we use relatively small amount of gas or just a few numbers of atoms or
molecules. It is possible to determine the mass of individual atoms or molecules, but
the numbers are messy, a gram is an incredibly large mass compared to the mass of an
atom. So we define a new mass unit, called an atomic mass unit or amu.
An atomic mass unit is defined as 1/12th the mass of a carbon-12 atom. Basically the
mass of a proton or neutron, or a proton or neutron have mass 1 amu. Note: Neutrons
and protons do not have exactly the same mass, but its close enough. Thus one
carbon-12 atom has mass 12 amu.
Helium-4 has an mass 4 amu or approximately and fluorine-19 has a mass of 19 amu
or approximately 1.99×10−23g. Which numbers would you rather use?
It would be a rare find to find a scale that measures in amu. So how do we relate amu
to grams? Here we define the concept of a mole. A mole (mol) is the number of
particles in a sample such that the mass in grams of the sample is equal to the mass in
amu of a single particle. So 1 mol of carbon-12 atoms has a mass of 12 g, 1 mol of
helium-4 has a mass of 4 g.
From this we can define the concept of molar mass, this is simply the grams per mole.
Carbon-12’s molar mass is 12g⋅mol−1.
3.3.3 Define the Avogadro constant
One day some body got bored and wondered exactly how many particles are in a mole,
this guy’s name was Avogadro (written in symbol form NA). I don’t think he actually
made a good measurement of his number, but hey he got a symbol named after him,
better than I could do.
(9)
NA=NumberofParticles / NumberofMoles
This number is called Avogadro’s number of Avogadro’s constant. The accepted value
is:
(10)
NA=6.023×(10^2)3
3.3.4 State that the equation of state of an ideal gas is
The number of moles of gas to the mass of gas is related by the molecular mass of the
gas. Therefore we can write the mass as:
(11)
m=nM
Where m is the mass, n is the number of moles and M is the molecular mass. If we
substitute this expression in to the general gas law:
3.3.1 State the macroscopic gas laws relating pressure, volume and temperature.
Boyle’s Law – For a gas at a constant temperature the volume and the pressure are
inversely proportional:
(12)
p∝1V
Plotting pressure vs. volume produces the
Plotting pressure vs. inverse volume
following graph:
produces a straight line graph:
If a thermodynamic system changes, but its temperature remains constant then the
following the initial pressure and volume are related to the final pressure and volume
by the following equation:
(13)
P1V1=P2V2
Charles’ Law – If the mass and pressure of a gas are held constant, the volume of the
gas is directly proportional to its absolute temperature.
(14)
V∝T
If a thermodynamic system changes, but its mass and pressure are held constant the
initial volume and temperature are related to the final volume and temperature by the
following equation:
(15)
V1 / T1=V2 / T2
Gay-Lussac’s Law ¬– If the volume of a sample of gas remains constant, the absolute
pressure of the gas is directly proportional to its absolute temperature.
(16)
P∝T
If a thermodynamic system changes, but the volume are held constant the initial
pressure and temperature are related to the final pressure and temperature by the
following equation:
(17)
P1/T1=P2/T2
All of the above mentioned laws required special circumstances that are not often
satisfied. A more general relationship between the volume, pressure and temperature
can be found:
(18)
P1V1/T1=P2V2/T2
This still does not take into account a change in mass. If the temperature and pressure
of a gas is held constant and more gas is added (mass increases) the volume must
increase. If the volume and temperature of a gas are held constant while more gas is
added the pressure must increase. If we combine these experimental observations we
can write an even more general relation:
(19)
P1V1/n1MT1=P2V2/n2MT2
Assuming the molecular mass is constant:
(20)
P1V1/n1T1=P2V2/n2T2
This can be written yet again as:
(21)
P1V1/n1T1=R
Or as the ideal gas law:
(22)
pV=nRT
An ideal gas being a gas that obeys the ideal gas law at all pressures, volumes and
temperatures. No real gas is ideal, but most gases obey the ideal gas law for low
pressure and high temperatures (above their freezing point).
R is called the universal gas constant, the value of R is dependent on the choice of
units:
(23)
R=8.314J⋅(mol^−1)⋅(K^−1)
The universal gas constant is given in the IB formula booklet.
3.3.5 Describe the concept of absolute zero and the Kelvin scale
If the volume of a gas is measured at constant pressure but at different temperatures a
linear relationship is found. If the line is extended to the left it will eventually intercept
the temperature axis, if the line went farther than this it would represent a negative
volume which does not make physical sense. If lines are plotted for several different
gases it can be found that they have different slopes, but intercept the temperature
axis at the same point. This temperature is in theory a minimum temperature, or
absolute zero. Absolute zero does not mean there is no internal energy, but only that
internal energy is at a minimum.
It should be noticed also that the equations presented above would provide
nonsensical answers for negative temperatures, i.e. negative volume or negative
pressure. This is simply a mathematical issue. Due to this, a new scale the Kelvin scale
was developed where 0 K is absolute zero. One degree Kelvin is the same change in
temperature as one degree Celsius. 0 K is defined as -273.16°C, for the purposes of
the IB -273°C is sufficient.
3.3.6 Solve problems using the equation of state of an ideal gas
3.3.7 Describe the kinetic model of an ideal gas
3.3.8 Explain the macroscopic behavior of an ideal gas in terms of molecular model
Straight from Physics 2nd Edition, Kerr, Kerr, Ruth pages 220-221:
The assumptions or postulates of the moving particle theory are extended for an ideal
gas to include:
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Gases consist of tiny particles called atoms (monatomic gases such as
neon and argon) or molecules.
The total number of molecules in any sample of gas is extremely large.
The molecules are in constant random motion.
The range of the intermolecular forces is small compared to the average
separation of the molecules.
The size of the particles is relatively small compared with the distance
between them.
Collisions of short duration occur between molecules and the walls of the
container and the collisions are perfectly elastic.
No forces act between particles except when they collide, and hence
particles move in straight lines.
Between collisions the molecules, obey Newton’s Laws of motion
The view of an ideal gas is one of molecules moving in random straight line paths at
constant speeds until they collide with the sides of the container or with one another.
Their paths over time are therefore zig-zags. Because the gas molecules can move
freely and are relatively far apart, they occupy the total volume of a container.
The large number of particles ensures that the number of particles moving in all
directions is constant at any time.
The pressure that the molecules exert is due to their collisions with the sides of the
container. As the temperature of a gas is increased, the average kinetic energy per
molecule increases. The increase in velocity of the molecules leads to a greater rate of
collisions, and each collision involves greater impulse. Hence the pressure of the gas
increases as the collisions with the sides of the container increase. When a force is
applied to a piston in a cylinder containing a volume of gas, the molecules take up a
smaller volume and hence collisions are more frequent leading to an increase in
pressure.
Because the collisions are perfectly elastic there is no loss in kinetic energy as a result
of the collisions. Temperature is a measure of the average kinetic energy per molecule.
Source: http://ibphysicsstuff.wikidot.com/ideal-gases