1. Unitary rates, unit analysis and scientific notation
The Mathematics of Chemistry 
Unitary rates, unit analysis and scientific notation

2. In your everyday lives and in science, unitary rates are used
In your everyday lives and in science, unitary rates are used. For example:
Apples at $5/ bag
Gasoline at $1.190 per Litre
A rate always compares a quantity of something to "one" thing of another.
In a unitary rate, the second quantity is always one (1).
E.G. 50 km/ h 23L/ min
$20/ kg 43 g/ mL 1 1 1 1

3. (Important note!)
Unitary rates can be “inverted” (i.e. flipped over) depending on the information that you start with...
When you go to the gas station the cost of gas is $1.08/ L...
This is because “most” people START BY KNOWING how much gas their car can take and WANT TO KNOW how much they will pay...
But the cost of gas could also be written as: 1L /$1.08 or 0.926L/$ ... This would be useful if you START BY KNOWING how much money you have and WANT TO KNOW how much gas you can get...

4. When two quantities are compared to each other and are found to change by the same ratio, then they are proportional to each other.
e.g. If the number of oranges doubles and the mass of the oranges also doubles the quantities are proportional.

5. In other words they produce a constant unitary rate...
Quantities that are proportional to each other always produce a constant value when one of the quantities is divided by the other.
In other words they produce a constant unitary rate...
No. of Rice Grains
Mass (g)
400
7.0
200
3.5
100
1.75
Note: The ratio of grains/mass is constant, therefore the quantities are proportional !

6. Eg. What is the density of a 45.6 g object if its volume is 10.2 mL?
Unitary rates are used in many calculations in chemistry and physics.
Eg. What is the density of a 45.6 g object if its volume is mL?

7. If a car travels 100km/h, how far will the car go in 4 hours?
100km x 4 hr hr = 400 km
Notice what happens to the units in the calculations!!
“Unit Analysis” involves the elimination of quantities (units) common to the numerator and the denominator of a calculation until the correct unit is derived.

8. initial amount(single) x unitary rate
 Unknown amount =
initial amount(single) x unitary rate
Ex. If a car can go 80 km in 1 hr, how far can it go in 8.5 hr? # Km = 8.5hr x 80km hr #Km = 680 Km

9. E.g. Light travels 300 000 km/s. How far will light travel in 1us
(Give your answer in km, m, and mm)

10. A unit factor compares two quantities using a ratio except the denominator is not one (1).
Eg.: 25 L / 100km
$10 / 12 donuts

11. Eg.: Unitary Rates = 50 peas/ 16.09 g How many peas are in 1000g?
Eg.: Unitary Rates = 50 peas/ g
How many peas are in 1000g?
Could you lift one million peas off the ground? Prove your answer.
Unknown amount = initial amount X unitary rate

12. How about a double Unit conversion???
If eggs are $1.44/dozen, and there are 12 eggs per dozen, how many eggs can be bought for $4.32?
Unknown Amount = Initial amount x Unitary rate
(“what you start” with x rate)
# eggs = $4.32 x 1 dozen x 12 eggs
$ dozen
# Eggs = 36

13. The gas tank of a Canadian car holds 39. 5L of gas
The gas tank of a Canadian car holds 39.5L of gas. If 1 L of gas is equal to US gallons, and gas is $1.26/gal in Texas…. How much will it cost to fill up the gas tank in Texas?
Unknown Amount = Initial amount x Unitary rate
# of $ = L x gal x $ 1.26
1 L gal
= $13.10

14. Practice Time: Hebden Pages
Pg 11 # 1 a,c,e,g
Pg 14 # 2 a,c,e,g,I
Pg 15 # 3,5,6,7,9,10

15. SCIENTIFIC NOTATION (See pg.30-32)
Since chemists deal with very large and very small numbers, they must have a way of working with these kinds of numbers.
The method that they use is called scientific notation
To write a number in scientific notation, the number is expressed as a number between one (1) and 10, then multiplied by 10 raised to a certain power.
e.g. 6.02× ×10-5
  mantissa

16. Decimal to scientific notation
Count the number of times you move the decimal point to produce a number between one (1) and ten (10)
That number times ten (10) to the number of decimal movements
Note: If the “real” # is bigger than your decimal, the exponent is positive 

17. Scientific to decimal notation
Write the mantissa; move the decimal point the number of places indicated by the exponent.
The decimal point is moved to the right if the exponent is positive
The decimal point is moved to the left if the exponent is negative   

18. Multiplying/dividing exponential numbers
Record these examples, then formulate a rule. 
RULE!!!!
Do the operation ( multiply or divide the mantissa as asked) Then…
If multiplying– always ADD the exponents in final answer.
If dividing– always SUBTRACT the exponents in final answer.

19. Adding/subtracting exponential numbers
Record the example, then formulate a rule.  
RULE!!!
1) Change the numbers so that they are multiplied by ten to the same exponent
2) Once the exponents are the same, add or subtract the resultant numbers (leave exponent as is!)

20. Do the Questions indicated on the worksheet