orders of magnitude and units. the mole: - the amount of a substance can be described using moles. -...
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Orders of Magnitude and Units
The ‘mole’:
- The amount of a substance can be described using ‘moles’.
- ‘One mole’ of a substance has 6 x 1023 molecules in it. (This number is called the Avogadro constant)
- So a chemist may measure out 3 moles of sulphur and she would know that she has 18 x 1023
molecules of sulphur.
1.1 The Realm of Physics
Q1. How many molecules are there in the Sun?
Info: - Mass of Sun = 1030 kg- Assume it is 100% Hydrogen- Avogadro constant = No. of molecules in one mole of a substance = 6 x 1023
- Mass of one mole of Hydrogen = 2g
A. Mass of Sun = 1030 x 1000 = 1033 gNo. of moles of Hydrogen in Sun = 1033 / 2 = 5 x 1032
No. of molecules in Sun = ( 6 x 1023 ) x ( 5 x 1032 )= 3 x 1056 molecules
Orders of Magnitude
Orders of magnitude are numbers on a scale where each number is rounded to the nearest power of ten. This allows us to compare measurements, sizes etc.
E.g. A giraffe is about 6m tall. So to the nearest power of ten we can say it is 10m = 1x101m = 101m tall.
An ant is about 0.7mm tall. So to the nearest power of ten we can say it is 1mm = 1x10-3m = 10-3m tall.
So if an ant is 10-3m tall and the giraffe 101m tall, then the giraffe is bigger by four orders of magnitude.
Orders of magnitude link
Order of Magnitude of some Masses Order of Magnitude of some
Lengths
MASS grams LENGTH meters
electron 10-27 diameter of nucleus 10-15
proton 10-24 diameter of atom 10-10
virus 10-16 radius of virus 10-7
amoeba 10-5 radius of amoeba 10-4
raindrop 10-3 height of human being 100
ant 100 radius of earth 107
human being 105 radius of sun 109
pyramid 1013 earth-sun distance 1011
earth 1027 radius of solar system 1013
sun 1033 distance of sun to nearest star
1016
milky way galaxy 1044 radius of milky way galaxy 1021
the Universe 1055 radius of visible Universe 1026
Q2. There are about 1x1028 molecules of air in the lab. So by how many orders of magnitude are there more molecules in the Sun than in the lab?
A. 1056 / 1028 = 1028 so 28 orders of magnitude more molecules in the Sun.
Q3. Determine the ratio of the diameter of a hydrogen atom to the diameter of a hydrogen nucleus to the nearest order of magnitude.
A. Ratio = 1015 / 1010 = 105
Prefixes
Power Prefix Symbol
1015 peta P
1012 terra T
109 giga G
106 mega M
103 kilo k
Power Prefix Symbol
10-15 femto f
10-12 pico p
10-9 nano n
10-6 micro µ
10-3 milli m
Quantities and Units
A physical quantity is a measurable feature of an item or substance.
A physical quantity will have a value and usually a unit. (Note: Some quantities such as ‘strain’ are dimensionless and have no unit).
E.g. A current of 5.3A ; A mass of 1.5x108kg
Base quantities
The SI system of units starts with seven base quantities. All other quantities are derived from these.
Base quantity Base Unit AbbreviationBase quantity Base Unit Abbreviation
mass (m)
Length (l)
time (t)
temperature (T)
electric current (I)
amount of substance (n)
luminous intensity (Iv)
Base quantity Base Unit Abbreviation
mass (m) kilogram kg
Length (l) metre m
time (t) second s
temperature (T) Kelvin K
electric current (I) Ampere A
amount of substance (n) mole mol
luminous intensity (Iv) candela cd
Derived units
The seven base units were defined arbitrarily. The sizes of all other units are derived from base units.
E.g. Charge in coulombsThis comes from : Charge = Current x time
so… coulombs = amps x secondsor… C = A x sso… C could be written in base
units as As (amp seconds)
Homogeneity
If the units of both side of an equation can be proved to be the same, we say it is dimensionally homogeneous.
E.g. Velocity = Frequency x wavelengthms-1 = s-1 x mms-1 = ms-1
homogeneous, therefore this formula is correct.
Dimensional Analysis
The dimensions of a physical quantity show how it is related to base quantities.
Dimensional homogeneity and a bit of guesswork can be used to prove simple equations.
E.g. Experimental work suggests that the period of oscillation of a pendulum moving through small angles depends upon its length, mass and the gravitational field strength, g.
So we can write Period = k mx ly gz
Where k is a dimensionless constant and x,y and z are unknown numbers.
So… s = kgx my (ms-2)z
s1 = kgx my+z s-2z
Now equate both sides of the equation:
For s 1 = -2z so z = -1/2
For kg 0 = x
For m 0 = y+z so y = +1/2
So… Period = k m0 l1/2 g-1/2
Or… Period = k l
g
Q.
Consider a sphere (radius, r) moving through a fluid of viscosity η at velocity v.
Experimental work suggests that the force acting upon it is related to these quantities. Use dimensional analysis to determine the formula.
(Note: the units of viscosity are Nsm-2)
A. You should prove… F = k ηrv
(F = 6π ηrv)
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