Tiny Numbers, Sizes and Units
This chapter uses some unusual numbers. The charge of an electron is 1.6 × 10−19 coulomb, its mass is 9.1 × 10−31 kg, and its charge-to-mass ratio is 1.76 × 1011 C/kg. A proton is about 1837 times heavier than an electron, and a shell can hold 2n2 electrons. This page shows you how to read and use such numbers: powers of ten, standard form, the units kg, coulomb and u, squares, and ratios. It also gives you a feel for how unbelievably small an atom and its nucleus (ന്യൂക്ലിയസ്) are.
- Multiplying and dividing by 10, 100, 1000.
- Decimals such as 0.1 and 0.01. Nothing else – every idea is built up here.
1. Powers of ten
Writing 10 × 10 × 10 is slow, so we write 103 (“ten to the power three”). The small raised number, the power or exponent, tells how many tens are multiplied. A negative power means dividing by tens: 10−3 = 1 ÷ 103 = 1/1000 = 0.001.
| Power | 103 | 102 | 101 | 100 | 10−1 | 10−2 | 10−3 |
|---|---|---|---|---|---|---|---|
| Number | 1000 | 100 | 10 | 1 | 0.1 | 0.01 | 0.001 |
| Meaning | 10 × 10 × 10 | 10 × 10 | 10 | one | 1/10 | 1/100 | 1/1000 |
Two patterns to remember:
- 10n (n positive) is 1 followed by n zeros: 106 = 1 000 000 (a million).
- 10−n is a decimal with the 1 in the n-th place after the point: 10−4 = 0.0001 (the 1 is in the 4th place).
Slide to the powers marked on the line. They are numbers you will meet in this chapter. Each step to the left makes the number 10 times smaller.
2. Standard form – writing very big and very small numbers
A number is in standard form (scientific notation) when it is written as
So 1.6 × 10−19 is in standard form, and so is 9.1 × 10−31. But 16 × 10−20 is not (16 is bigger than 10), and neither is 0.91 × 10−30 (0.91 is less than 1). Read 1.6 × 10−19 aloud as “one point six times ten to the power minus nineteen”.
How to convert: move the decimal point until exactly one non-zero digit is in front of it, and count the hops.
- Point moved left (big number) → the power is positive.
- Point moved right (small number) → the power is negative.
Example 1 · A big number: 176 000 000 000
Example 2 · A tiny number: the charge of an electron
Multiplying and dividing powers of ten
For example 103 × 102 = 105 (1000 × 100 = 100 000), and 10−19 ÷ 1011 = 10−19 − 11 = 10−30. To multiply or divide numbers in standard form, deal with the front numbers and the powers separately.
Worked example · The mass of an electron (a preview of page 4)
Q. J. J. Thomson measured the charge-to-mass ratio of the electron: e/m = 1.76 × 1011 C/kg. Millikan measured its charge: e = 1.6 × 10−19 C. Find the mass m of the electron. (m = e ÷ (e/m).)
Divide the front numbers: 1.6 ÷ 1.76. Subtract the powers: −19 − 11. Then put the answer back into standard form.
3. How small is an atom?
Atoms are far too small to see, even with the best ordinary microscope. Sizes in this lab are given in metres (m) and are approximate. Real atoms of different elements range from about 1 to 3 × 10−10 m across. The nucleus (ന്യൂക്ലിയസ്) at the centre of an atom is smaller still.
Each step down the ladder is a huge jump in size. Going from a cell to an atom is a jump of about 500 000 times, and from an atom to its nucleus about 10 000 to 100 000 times. These values are approximate.
Try it. A human hair is about 0.1 mm = 1 × 10−4 m thick. An atom is about 1 × 10−10 m across. About how many atoms would fit side by side across the hair?
4. Units used in this chapter
| Quantity | Unit | Symbol | Where you will meet it |
|---|---|---|---|
| Length, size | metre | m | sizes of atoms (about 10−10 m) |
| Mass (മാസ്) | kilogram | kg | mass of the electron, 9.1 × 10−31 kg (page 4) |
| Charge (ചാർജ്) | coulomb (കൂളോം (C)) | C | charge of the electron, 1.6 × 10−19 C (page 4) |
| Charge ÷ mass | coulomb per kilogram | C/kg | e/m അനുപാതം of the electron, 1.76 × 1011 C/kg (page 4) |
| Mass of atoms and particles | unified atomic mass unit (യൂണിഫൈഡ് അറ്റോമിക് മാസ് യൂണിറ്റ് (u)) | u | proton 1.00727 u, neutron 1.00866 u, electron 0.000548 u (page 7) |
Why a special unit for atoms? A proton has a mass of about 0.000 000 000 000 000 000 000 000 001 67 kg (1.67 × 10−27 kg). Writing that all the time would be painful! So chemists use the unified atomic mass unit, u. One u is about 1.66 × 10−27 kg (go further), so a proton is about 1 u and a neutron is also about 1 u. Your textbook says: the unified atomic mass unit (u) is the unit used to express the masses of atoms.
Go further. The charge of one electron is 1.6 × 10−19 C. How many electrons together carry a charge of 1 coulomb?
5. Squares and the 2n² table
The square of a number is the number multiplied by itself: n2 = n × n, so 32 = 9. On page 9 you will learn that the n-th shell of an atom can hold at most 2n2 electrons. That means: square n first, then double it. (Not (2n)2, which would square the 2 as well!)
| n (shell number) | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| shell name | K | L | M | N |
| n2 | 1 | 4 | 9 | 16 |
| 2n2 | 2 | 8 | 18 | 32 |
6. Ratios and the fraction 1/1837
A ratio compares two amounts. If a proton is 1837 times as heavy as an electron, the ratio of their masses is 1837 : 1. Turned around, the mass of an electron is 1/1837 of the mass of a proton. That is the same as dividing by 1837: 1 ÷ 1837 ≈ 0.000544. Your textbook gives this fraction on page 14. Using the masses in u: 1.00727 ÷ 1837 ≈ 0.000548 u – the electron's mass in Table 1.2.
An imaginary balance: real particles can't be weighed like this. The textbook value is 1837; with the masses in Table 1.2 (1.00727 u ÷ 0.000548 u) you get about 1838, so the two agree.
- 10n = 1 followed by n zeros; 10−n = 1 ÷ 10n, a decimal with the 1 in the n-th place.
- Standard form: a × 10n with 1 ≤ a < 10. Point moved left → positive power; moved right → negative power.
- 10a × 10b = 10a + b; 10a ÷ 10b = 10a − b.
- An atom is about 10−10 m across; its nucleus is 10 000 to 100 000 times smaller (approx.) – like a pea in a cricket ground.
- Units: kg (mass), coulomb C (charge), u (masses of atoms; proton ≈ neutron ≈ 1 u).
- 2n2 for n = 1, 2, 3, 4 gives 2, 8, 18, 32. An electron's mass is 1/1837 of a proton's.
7. Check yourself
Next: Inside the atom: the discharge tube and the electron – the chapter begins ▶