Learning aids

Significant Figures (ground rules):

In a number, not all digits matter. For example, I could say the human population on Terra is 7-billion (7,000,000) without being wrong. I do not need to care if it is really 7,000,002, for the difference of 2 people does not make a significant inpact on the number 7-billion. Hence, the 2 would be insignificant, and I should now explain how chemists/physisists use the term. Just as a side note, please know that the term 'digit' and 'figure' are synonomous when talking about significance.

Non-zero digits are always significant. A number like 123. would have 3 digits; 1.867 has 4 digits, and 7,000,002. has 7 digits. This last one may seem to contridict my introduction, but I shall explain why it doesn't later.

Zeros that procede a number are always insignificant. A number like 0.02 would have one digit that is significant, and a number like 0.000000000002 would also only have one significant figure.

'Nested' zeros are always significant. The number 0.202 has 3 significant digits, as would the number 1.02, or the number 301.

Trailing zeros are sometimes significant. If there is a decimal point, the zeros are significant. If there is no decimal point, they are not significant. For example, the number 1000 has one digit, while the numbers 1000. and 1.000 have 4 each.

Now, most people will complain about significance of various zeros. This boils down to the instrument used to get the measurement.

In my opening example, I am taking a rough estimate from the web. I may be more accurate to say that there are 7.1 billion people, but many countries do not keep accurate population statistics, and as such, there is some doubt as the exact number. This implys that there could be anywhere between 7.0 and 7.2 billion people.

If I said that there were 7,000,001. people in the world, then I would be a flat out liar, as that number implies that there are anywhere between 7,000,000. and 7,000,002 people in the world. Based on ths same arguement above, there is no way I can state the number of people in the world down to the exact person.

Let me try another example. If I was looking at the temperature on a thermometer on a day in late March, lets say that it says 100.0ºF as the temperature. If I told someone this, they will know that the temperature is between 101.1ºF and 99.9ºF, and as such, there is no major discrepencies.

Let us take the same day, but this time, I shall tell you that the thermometer says the temperature is 100.ºF. You know, by me saying that, that the temperature is between 101.0ºF and 99.0ºF as once again the last significant figure is in doubt. However, for a difference of 2 degrees, you should still be okay.

Now, I get lazy. Lets once again take the same day in the March and I will tell you that the thermometer measured the temperature to be 100ºF. According to the number I gave you, the temperature could now be between 200.0ºF and 0.0ºF, since the last significant digit is always in doubt.

Now, ignore the temperature I said just a second. We assumed we are in March, and as such, a blizzard is just as likely as a heatwave. So, let us say we have that blizzard, and it is 10.ºF out. I can still tell everyone it is 100ºF with a clean conscience as 10.0ºF is within 200.º and 0.ºF.

The moral of the story? Make sure you report all significant figures as they are given, unless you'd rather have a bikiniclad-lynch-mob after you in a foot of snow!

Significant Figures (their use with operations):

Addition/subraction:

When you add or subtact numbers, you keep the location of the weakest significant figures of the weakest measurement. Note, it is possible to 'gain' or 'lose' significant figures if digits add up in certain ways. Example #2 shows an instance where you can 'loose' a figure and example #3 shows an instance where you can 'gain' a figure.

Examples (numbers in brackets are the number of significant figures a value has.)
	1) Add: 2.00 + 2.34 4.34 = 4.	[1] [3] [1]	2) Subtract: 115.31600 - 112.85723 2.45877 = 2.459	[6] [8] [4]	3) Add: 0.316 + 9.888 10.204 = 10.204	[3] [4] [5]

Multipication/division:

When you multiply or divide numbers, you keep the number of the weakest significant figures of the weakest measurement. Note, because of this, it is not possible to 'gain' figures, but it is not possible to 'lose' them, reguardless of how simple they are, example #4 shows this well.

Examples (numbers in brackets are the number of significant figures a value has.)
	1) Multiply: 2.00 x 2.34 4.68 = 4.	[1] [3] [1]	2) Divide: 115.31600 ÷ 112.85723 1.0217865... = 1.02179	[6] [8] [6]	3) Multiply: 0.316 x 9.888 3.1246... = 3.12	[3] [4] [3]	4) Divide: 20.0 ÷ 10.0 2.00 = 2.00	[3] [3] [3]

Mole Relationships:

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Equation

Terms therein

Sample Question

Solution with Explanation

µ = m·n^-1

µ = Molar Mass
m = Mass
n = Moles

You are given 153-mole sample of Oxygen. What is the mass of the sample of Oxygen (in Grams) if the molar mass of Oxygen is 31.9988 g·mol^-1?

1) Pick the apropriate formula from the chart.

m·n^-1

2) Re-solve formula for desired term.
(Mass [m] in this case.)

n·µ

3) Plug-in known values into the appropriate varibles.

(153 mol)·(31.9988 g·mol^-1)

4) Cancel any units possible
(In this case, mol·mol^-1.)

(153 ~~mol~~)·(31.9988 g·~~mol^-1~~)

5) Rewrite problem without the canceled units.
(This step can be skipped with practice.)

(153)·(31.9988 g)

6) Carry out multiplications presented, carry all unis not canceled out in step 4, and then underline significant figure.

485.8164 g

7) Write out and box final answer after rounding to correct significant figures and do not forget the unit!

486. g

P·V = n·R·T

P = Pressure
V = Volume
n = Moles
R = Universal Gas Constant
T = Temperature

What is the volume (in Liters) of a 1.00-mole sample of ideal gas at STP? (IE: 1.00 ATM of pressure and at temperature of 273.15 K)

1) Pick the apropriate formula from the chart.

P·V

n·R·T

2) Re-solve formula for desired term.
(Volume [V] in this case.)

n·R·T·P^-1

3) Plug-in known values into the appropriate varibles.

(1.00 mol)·(0.082058 L·ATM·K^-1·mol^-1)·(273.15 K)·(1.00 ATM^-1)

4) Cancel any units possible
(In this case, mol·mol^-1, ATM·ATM^-1, and K·K^-1.)

(1.00 ~~mol~~)·(0.082058 L·~~ATM~~·~~K^-1~~·~~mol^-1~~)·(273.15 K)·(1.00 ~~ATM^-1~~)

5) Rewrite problem without the canceled units.
(This step can be skipped with practice.)

(1.00)·(0.082058 L)·(273.15)·(1.00)

6) Carry out multiplications presented, carry all unis not canceled out in step 4, and then underline significant figure.

22.4141427 L

7) Write out and box final answer after rounding to correct significant figures and do not forget the unit!

22.414 L

M = n·V^-1

M = Molarity
n = Moles
V = volume

You need to titrate 1.00-mole of Sodium Hydroxide (NaOH) with exactly 1.00-mole of Hydrogen ions (H⁺). You are given a 15.0M solution of Hydrochloric acid (HCl). How many liters do you need to deliver of the HCl to neutralize the NaOH?

1) Pick the apropriate formula from the chart.

n·V^-1

2) Re-solve formula for desired term.
(Volume [V] in this case.)

n·M^-1

3) Plug-in known values into the appropriate varibles.

(1.00 mol)·((15.0 mol)·(1.00 L)^-1)^-1

3.5) To prevent mistakes, invert any fractions that are to the -1 power.

(1.00 mol)·(1.00 L)·(15.0 mol)^-1

4) Cancel any units possible
(In this case, mol·mol^-1.)

(1.00 ~~mol~~)·(1.00 L)·(15.0 ~~mol~~)^-1

5) Rewrite problem without the canceled units.
(This step can be skipped with practice.)

(1.00)·(1.00 L)·(15.0^-1)

6) Carry out multiplications presented, carry all unis not canceled out in step 4, and then underline significant figure.

(0.06666[r] L)

7) Write out and box final answer after rounding to correct significant figures and do not forget the unit!

0.0667 L

n = N·NA^-1

n = Moles
N = Number of Particles
NA = Avagadro's Number

If you went to the store and brought 2.39x10^-22 moles of eggs to the counter, how many dozens of eggs are you buying?

1) Pick the apropriate formula from the chart.

N·NA^-1

2) Re-solve formula for desired term.
(Volume [V] in this case.)

n·NA

3) Plug-in known values into the appropriate varibles.

(2.39x10^-22 mol)·(6.022x10²³ eggs·mol^-1)

4) Cancel any units possible
(In this case, mol·mol^-1.)

(2.39x10^-22 ~~mol~~)·(6.022x10²³ eggs·~~mol^-1~~)

5) Rewrite problem without the canceled units.
(This step can be skipped with practice.)

(2.39x10^-22)·(6.022x10²³ eggs)

6) Carry out multiplications presented, carry all unis not canceled out in step 4, and then underline significant figure.

143.9258 eggs

6.5) We found the number of eggs, we wanted the number of dozens, so dont forget to convert. There are 12 eggs to a dozen.

(143.9258 eggs)·((1 doz)·(12 eggs)^-1) = 11.99 doz

7) Write out and box final answer after rounding to correct significant figures and do not forget the unit!

12.0 doz

Significant Figures (ground rules):

Significant Figures (their use with operations):

Physical Constants:

Mole Relationships:

IR/P-NMR shifts for Organic Chemistry: