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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