The Ideal Gas Laws

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					The Ideal Gas Laws


   Chapter 14
Expectations

After this chapter, students will:
Know what a “mole” is
Understand and apply atomic mass, the atomic
  mass unit, and Avogadro’s number
Understand how an ideal gas differs from real
  ones
Use the ideal gas equation, Boyle’s Law, and
  Charles’ Law, to solve problems
Expectations

After this chapter, students will:
understand the connection between the
  macroscopic properties of gases and the
  microscopic mechanics of gas molecules
Preliminaries: the Mole

A mole is a very large number of discrete objects,
  such as atoms, molecules, or sand grains.

Specifically, it is Avogadro’s Number (NA) of such
  things: 6.022×1023 of them.

The mole (“mol”) is not a dimensional unit; it is a
  label.
Amadeo Avogadro

1776 – 1856
Native of Turin, Italy

Hypothesized that equal volumes
of gases at the same temperature
and pressure contained equal
numbers of molecules.
(He was correct, too.)
The Mole and Atomic Mass

Mathematical definition: 12 g of C12 contains one
 mole of carbon-12 atoms.
                            12 g   .012 kg
Mass of one   C12   atom:                     1.993 10 -26 kg
                             N A 6.022 10 23
The mass of one C12 atom is also 12 atomic mass
  units (amu), so:
               1.993 10 -26 kg
       1 amu                    1.661 10 -27 kg
                     12
The Mole and Atomic Mass

Atomic masses for the elements may be found in the
  periodic table of the elements, located inside the
  back cover of your textbook.

These are often erroneously called “atomic
  weights.”

Atomic masses may be added to calculate molecular
  masses for chemical compounds (or diatomic
  elements).
The Mole: Calculations

If we have N particles, how many moles is that?
                      N
 number of moles
                   n
                      NA
If we have a given mass of something, how many
   moles do we have?
       mass of sample       mass of sample
    n                
       mass per mole    atomic or molecular mass
The Ideal Gas

The notion of an “ideal” gas developed from the
  efforts of scientists in the 18th and 19th centuries
  to link the macroscopic behavior of gases
  (volume, temperature, and pressure) to the
  Newtonian mechanics of the tiny particles that
  were increasingly seen as the microscopic
  constituents of gases.
The Ideal Gas

An ideal gas was one whose particles are well-
 behaved, in terms of the Newtonian theory of
 collisions: elastic collisions and the impulse-
 momentum theorem.

An ideal gas is one in which the particles have no
 interaction, except for perfectly-elastic collisions
 with each other, and with the walls of their
 container.
The Ideal Gas

An ideal gas has no chemistry. That is, the particles
 (atoms or molecules) have no tendency to “stick”
 to other particles through chemical bonds.

Inert gases (He, Ne, Ar, Kr, Xe, Rn) at low densities
  are very good approximations to the ideal gas.

Our analytic model of the ideal gas gives us insights
 into the properties of many real gases, inert or not.
The Ideal Gas Equation
Observations from experience

The pressure of a gas is directly proportional to the
  number of moles of particles in a given space.
  Example: blow up a balloon, and you’re adding to
  n, the number of moles of molecules.

Conclusion:    Pn
The Ideal Gas Equation
Observations from experience

The pressure of a gas is directly proportional to its
  temperature. Example: toss a spray can into a fire
  (no, wait, really, don’t do it, just think about it).
  Increasing pressure will cause the can to fail
  catastrophically.

Conclusion:     P T
The Ideal Gas Equation
Observations from experience

The pressure of a gas is inversely proportional to its
  volume. Example: squeeze the air in a half-filled
  balloon down to one end and squeeze it tighter.
  Increased pressure makes the balloon’s skin tight.

                   1
Conclusion:     P
                   V
The Ideal Gas Equation
Combine the observations
                    nT
                 P
                    V
A constant of proportionality, R, makes this an
  equation:

              nT
          PR    or PV  nRT
              V
The Ideal Gas Equation
The constant of proportionality, R, is called the
  universal gas constant. Its value and units
  depend on the units used for P, V, and T.
  pressure     volume
                                   absolute

                   PV  nRT
                                   temperature


                                universal gas constant
             number of moles


Value and SI units of R: 8.31 J / (mol K)
The Ideal Gas Equation
We can also write the ideal gas equation in terms of
 the number of particles, N, instead of the number
 of moles, n.
Since N = n·NA, we can both multiply and divide the
  right-hand side by NA:
              R 
   PV  nN A 
             N T             Boltzmann’s constant
              A
                      R
   PV  NkT where k      1.38 10-23 J/K
                      NA
Ludwig Boltzmann

Austrian physicist

1844 – 1906
Boyle’s Law

Suppose we hold both n and T constant: how are P
  and V related?
      PV  nRT               PV  constant


      PV1  P2V2
       1



This is called Boyle’s Law.
Robert Boyle

Irish mathematician

1627 – 1691
Charles’ Law

Suppose we hold both n and P constant: how are T
  and V related?
                        V nR
    PV  nRT                 constant
                        T   P

    V1 V2
      
    T1 T2
This is called Charles’ Law.
Jacques Alexandre Cesar Charles

French scientist

1746 – 1823

Built and flew the first
large hydrogen-filled
balloon.
Kinetic Theory of the Ideal Gas

Macroscopic properties of a gas: temperature,
 pressure, volume, density

Microscopic properties of the particles making up
 the gas: mass, velocity, momentum, kinetic
 energy

How are they related?
Kinetic Theory of the Ideal Gas
Consider a gas molecule contained in a cube having
 edge length L.
The molecule’s mass is m, and
its velocity (in the X direction
only) is v.
Time between collisions with the
right-hand wall:      2L
                    t
                         v
Kinetic Theory of the Ideal Gas
The time between collisions with the right-hand wall is
  just the round-trip time:    2L
                            t
                                v
From the impulse-momentum
theorem, we can calculate the
average force exerted on the
particle by the wall:


      J  F t  p  p f  p0  m v   mv
Kinetic Theory of the Ideal Gas
Substitute for the time and simplify:
J  F t  p  p f  p0  m v   mv
   2L
F     2mv
    v
     mv 2
F
      L

By Newton’s third law, the average          mv 2
force exerted on the wall is
                                         F
                                             L
Kinetic Theory of the Ideal Gas
The average force on the wall from one particle is
          mv 2
       F
           L
If there are N particles, and
their directions are random, we
could expect 1/3 of them to be
moving in the X direction.
                                         2
                                N mv
Total force on the wall:     F
                                3 L
Kinetic Theory of the Ideal Gas
Average pressure on the wall:
      N mv 2
   F         N mv 2
P   3 2L 
   A    L    3 L3

But   V L  3
                  So:

            2
   N mv               N
P               PV  mv 2  NkT
   3 V                3
Kinetic Theory of the Ideal Gas
Substituting kinetic energy:
     1    1 1    2  2
 kT  mv   2 mv   KE
        2

     3    3 2      3
      3
 KE  kT
      2
So, we see that for an ideal gas,
the average molecular kinetic energy
is directly proportional to the
absolute temperature.
Kinetic Theory of the Ideal Gas
     3
 KE  kT This result is true for any ideal gas.
     2
By a similar argument, if an ideal gas is monatomic
 (the gas particles are single atoms), the internal
 energy of n moles of the gas at an absolute
 temperature T is
                          3
                       U  nRT
                          2

				
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