Transcript Avogadro`s number

Recap: Why Atoms?
• Chemical combination rules (Dalton)
• Success of kinetic theory in describing behavior of matter
– Predictions that follow from the theory are confirmed, although
atoms are not “directly” observed
• Brownian motion
– A way of seeing atoms “directly”
– Again, predictions based on atomic theory are confirmed by
experiments
• X-ray diffraction
– Studied in detail in the early 20th century
Response Sheet Questions
• Are scientists and philosophers really at odds?
• Isn’t it true that science is, in part, a collection of facts?
Why would the Royal Society refuse to open the
notebooks in Popper’s fable?
• Who was Joanna Southcott?
• Relation between the arts and sciences, and mutual
influences?
How Big are They?
• Clearly, very small!
• Too small to be visible in the best optical microscopes
– Microscopes can only resolve an object that is comparable in size
or larger than the wavelength of the light used to illuminate it
– For visible light, the smallest structures that can be seen are about
400 nanometers, or 4  10–7 m
– Atoms must be smaller than this!
What is Light?
• Light is a type of wave
• Other common examples:
water waves, sound
• A wave is a disturbance
in a medium (water, air,
etc.) that propagates
• Typically the medium
itself does not move
much
Anatomy of a Wave
crest
wavelength
2 x amplitude
trough
direction of wave motion
Electromagnetic Waves
• Medium: the electric and magnetic field
• Speed = 3  105 km/sec (about 186,000 mi/sec)
The Electromagnetic Spectrum
Visible Light
• The color of visible light is
determined by its wavelength
• White light is a mixture of all
colors
• We can separate out individual
colors with a prism
Visible Light
400–440 nm
440–480 nm
480–530 nm
530–590 nm
590–630 nm
630–700 nm
Violet
Blue
Green
Yellow
Orange
Red
Longer wavelength
Shorter wavelength
So, How Big are They?
• Earliest estimate: Johann Loschmidt
(1865)
– Used results from kinetic theory to estimate
the size of an “air molecule”
• We no know there are several types of
molecules present in air
• They are roughly the same size, though!
– His result was about one millionth of a
millimeter
• In other words, about 10–3 m/106 or 10–9 m
• This is about 400 times smaller than the
smallest object visible in an optical
microscope
Brownian Motion
• Discovered in 1828 by Robert Brown,
a Scottish botanist
• He observed that microscopic pollen
grains suspended in a liquid move
around erratically, even though the
liquid itself has no observable motion
• Explanation: the grains are being
jostled and buffeted by unseen atoms
• The smaller the grain, the more
violently it is agitated
Size of Atoms
• In 1905, Einstein worked out several predictions regarding
Brownian motion using atomic theory
• Confirmed by Jean Perrin (1908)
– Nobel Prize for Physics 1926
• Based on his measurements, Perrin gave an accurate
estimate of the size of atoms: about 1-2  10–10 m
• The atomic scale is about a tenth of a nanometer
Making Predictions for
Brownian Motion
•
•
Question: How far does the grain move (on average) in
some time T?
Assumptions:
1. The grain is buffeted on all sides by unseen particles
2. Due to statistical fluctuations, there is occasionally an excess of
collisions on one side of the grain, leading to a slight “kick” in the
opposite direction
3. The direction of the kicks is random, i.e. the direction of one kick
is independent of any previous kicks
4. On average, these “kicks” occur every T0 seconds
5. On average the kicks are the same “size”, i.e. they cause the same
displacement of the grain
Model Calculation
•
•
•
•
“Random walk” (one dimensional, for simplicity)
Grain starts at square 0
Each step corresponds to a “kick”
For each step, flip a coin
– Heads, go right
– Tails, go left
• Do it many times and see how far the grain has gotten after
N steps, on average
0
How do we work it out?
• Could flip many coins, but easier to
let a computer do it!
– It never gets tired or bored
• So we let the computer
– Flip virtual coins and determine how far
the grain gets after N steps
– Repeat a large number of times
– Calculate the average distance traveled in
N steps
– Then repeat for some other value of N
• How does the average distance D
change with N?
The Result
• The average distance is proportional to the square root of N
D N
• Since each step took T0 seconds, the total time taken is
T  NT0
• Hence
D T
Confirmed by Perrin!
The Electromagnetic Spectrum
X-Ray Diffraction
• X-rays have wavelengths
comparable to atomic sizes
• We can “see” atoms and
molecules by bouncing Xrays off them
• Crystals and molecules
reflect X-rays in patterns
depending on their structures
• From the reflection pattern
one can figure out the
structure!
X-ray diffraction pattern of DNA
Avogadro’s Hypothesis
• Equal volumes of gases under the same conditions of
temperature and pressure have equal numbers of molecules
• Derived from the observations by Gay-Lussac and others
– gases unite in simple proportions by volume
– if a reaction of two gases produces a gas, the volume of gas
produced is also related by a simple proportion
• He also proposed that some gases (like oxygen and
hydrogen) are not made up of single atoms
• Why weren’t his idea quickly accepted?
– they indicated that Dalton’s atomic weights were wrong
– there was no agreement as to what a “molecule”
– he was not a particularly accomplished experimentalist
Loschmidt’s Number
• Avogadro’s Hypothesis predicts that one cubic
centimeter of any gas under standard conditions will
always contain the same number of molecules
• Avogadro, however, never calculated this number (he
had neither the experimental or theoretical
background to accomplish this)
• The first estimate of this quantity was made by
Loschmidt in 1865 from the kinetic theory of gases
• (Cannizzaro first promoted Avogadro’s ideas at the
Karlsruhe Conference of Chemists in 1860)
Avogadro’s Number
• Chemists prefer to use what we now call
“Avogadro’s Number” for many calculations
• Avogadro’s number is the number of oxygen
(O2) molecules in 32 grams of oxygen, or the
number of H2 molecules in 2 grams of H2
or the number of molecules per mole of molecules
• It is an honorary name (first used by Perrin in
1909) – still called Loschmidt’s Number in
Germany
• NA = 6.022 x 1023 (atoms or molecules) per mole
Estimating Avogadro’s Number