# Mathematical Relationships in Chemistry

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Mathematical Relationships in Chemistry

What You’ll Learn in this Unit
Significant Figures Scientific Notation Measurement Dimensional Analysis Error Density Graphical analysis

Review of Measurement Terms
Qualitative measurements - words Quantitative measurements – involves numbers (quantities) Depends on reliability of instrument Depends on care with which it is read

Precision vs. Accuracy Precision- the degree of agreement among several measurements of the same quantity. Accuracy- the agreement of a particular value with the true value

Uncertainty Basis for significant figures
All measurements are uncertain to some degree Random error - equal chance of being high or low- addressed by averaging measurements - expected

Significant Figures Meaningful digits in a measurement
The number of significant figures in your measurement will tell the reader how exact the instrumentation used If it is measured or estimated, it has sig figs. If not it is exact (e.g. 5 apples).

Significant Figures All numbers 1-9 are significant.
The problem are the ZEROS. Which ones count and which don’t? In between numbers 1-9 does Example: ……… has 4 sig figs Now let me tell you a story…

Left handed Archer There once was a left handed archer who loved to shoot decimals. Zeros could not stop his arrow but numbers could. If there is a decimal in the number begin on the left. Go through any zeros, come to the first number then all other numbers that follow are SIGNIFICANT! →0.0040 →50.401

No decimals If a number has no decimals you begin on the right hand side. Go through any zeros , come to the first number. Then all numbers after that count 5000← 405,000 ←

Doing the Math Multiplication and division, same number of sig figs in answer as the least in the problem Addition and subtraction, same number of decimal places in answer as least in problem.

Scientific Notation 100 = 1.0 x 102 0.001 = 1.0 x 10-3
-- This provides a way to show significant figures.

TOO QUICK FOR YOU! So here are the rules.. slowly!
Place decimal point after 1st real non-zero integer. (ex) 1.0 NOT 10.0 Raise 10 to the exponential which equals the number of places you moved.

Scientific Notation The product of 2.3 x 10 x 10 x 10 equals 2300 (2.3 x 103) Note: Moving the decimal to the left will increase the power of 10 Moving the decimal to the right will decrease the power of 10

Sample Problems 2387

Answers 2.387 x 103 7.031 X 10-5 2.9 x 109 8.900 X 10-3 9.01 X 107 2.10 X 10-6

Scientific Notation Multiplication and Division
Use of a calculator is permitted use it correctly No calculator? Multiply the coefficients, and add the exponents (3 x 104) x (2 x 102) = (2.1 x 103) x (4.0 x 10-7) = 6 x 106 8.4 x 10-4

Scientific Notation Multiplication and Division
In division, divide the coefficients, and subtract the exponent in the denominator from the numerator 3.0 x 105 6.0 x 102 5 x 102 =

Scientific Notation 7 x 10-8 3.17 x 10-5 (6.6 x 10-8) + (4.0 x 10-9) =
Addition and Subtraction Before numbers can be added or subtracted, the exponents must be the same Calculators will take care of this Doing it manually, you will have to make the exponents the same- it does not matter which one you change. (6.6 x 10-8) + (4.0 x 10-9) = (3.42 x 10-5) – (2.5 x 10-6) = 7 x 10-8 3.17 x 10-5

Measurement Every measurement has two parts
COMMON SI UNITS Symbol   Unit Name   Quantity   Definition    m meter length base unit    kg kilogram mass    s second time    K kelvin temperature       °C degree Celsius**   temperature    m3 cubic meter volume m3    L liter** dm3 = m3    N newton force kg·m/s2    J joule energy N·m    W watt power J/s    Pa pascal pressure N/m2    Hz hertz frequency 1/s Every measurement has two parts Number with the correct sig - figs Scale (unit) We use the Systeme Internationale (SI).

Metric Base Units Mass - kilogram (kg) Length- meter (m) Volume- (L)
Time - second (s) Temperature- Kelvin (K) Electric current- ampere (amp, A) Amount of substance- mole (mol) Energy – joule (j)

Prefixes giga- G 1,000,000,000 109 mega - M 1,000,000 106
kilo - k 1, deci- d centi- c milli- m micro- m nano- n

Using Units to solve problems
Dimensional Analysis Using Units to solve problems

Dimensional Analysis Use conversion factors to change the units
1 foot = 12 inches (equivalence statement) 12 in = = 1 ft ft in 2 conversion factors multiply by the one that will give you the correct units in your answer.

Example Problem The speed of light is 3.00 x 108 m/s. How far will a beam of light travel in 1.00 ns? Well, we know that 1.00 ns = 10-9 seconds (3.00 x 108 m) X (10-9 s) = x m/ns s (1.00 ns)

Example Problems 11 yards = 2 rod 40 rods = 1 furlong
8 furlongs = 1 mile The Kentucky Derby race is 1.25 miles. How long is the race in rods, furlongs, meters, and kilometers? A marathon race is 26 miles, 385 yards. What is this distance in rods, furlongs, meters, and kilometers?

Volume The space occupied by any sample of matter
Calculated for a solid by multiplying the length x width x height SI unit = cubic meter (m3) Everyday unit = Liter (L), which is non-SI

Units of Mass Mass is a measure of the quantity of matter
Weight is a force that measures the pull by gravity- it changes with location Mass is constant, regardless of location. The SI unit of mass is the kilogram (kg), even though a more convenient unit is the gram Measuring instrument is the balance scale

Density Which is heavier- lead or feathers? It depends upon the amount of the material A truckload of feathers is heavier than a small pellet of lead The relationship here is between mass and volume- called Density

Density Common units are g/mL, or possibly g/cm3, (or g/L for gas)
Ratio of mass to volume D = m/V Common units are g/mL, or possibly g/cm3, (or g/L for gas) Useful for identifying a compound Useful for predicting weight An intensive property- does not depend on what the material is

Things related to density
density of corn oil is 0.89 g/mL and water is 1.00 g/mL What happens when corn oil and water are mixed? Why? Will lead float?

Example Problem An empty container weighs g. Filled with carbon tetrachloride (density 1.53 g/cm3 ) the container weighs g. What is the volume of the container?

Density and Temperature
What happens to density as the temperature increases? Mass remains the same Most substances increase in volume as temperature increases Thus, density generally decreases as the temperature increases

Density and water Water is an important exception Over certain temperatures, the volume of water increases as the temperature decreases Does ice float in liquid water? Why?

Specific Gravity A comparison of the density of an object to a reference standard (which is usually water) at the same temperature Water density at 4 oC = 1 g/cm3

Formula Note there are no units left, since they cancel each other
D of substance (g/cm3) D of water (g/cm3) Note there are no units left, since they cancel each other Measured with a hydrometer Uses? Gem purity differentiating between different types of crude oils/gasoline urine tests for concentration of all chemicals in your urine Specific gravity =

Temperature Heat moves from warmer object to the cooler object Glass of iced tea gets colder? Remember that most substances expand with a temperature increase? Basis for thermometers

Temperature scales Celsius scale- named after a Swedish astronomer Uses the freezing point (0 oC) and boiling point (100 oC) of water as references Divided into 100 equal intervals, or degrees Celsius

Temperature scales Kelvin scale (or absolute scale) Named after Lord Kelvin K = oC + 273 A change of one degree Kelvin is the same as a change of one degree Celsius No degree sign is used

Temperature scales Water freezes at 273 K Water boils at 373 K 0 K is called absolute zero, and equals –273 oC

Temperature A measure of the average kinetic energy
Different temperature scales, all are talking about the same height of mercury. In lab take the reading in ºC then convert to our SI unit Kelvin ºC = K

100ºC = 212ºF 0ºC = 32ºF 100ºC = 180ºF 1ºC = (180/100)ºF 1ºC = 9/5ºF

Example problem A 55.0 gal drum weighs 75.0 lbs. when empty. What will the total mass be when filled with ethanol? density g/cm gal = L 1 lb = 454 g

Error Calculations X 100 accepted value
Error = Experimental value- accepted value % error = [error] accepted value X 100

Graphing The relationship between two variables is often determined by graphing A graph is a “picture” of the data

Graphing Rules – 10 items Plot the independent variable on the x-axis (abscissa) – the horizontal axis. Generally controlled by the experimenter Plot the dependent variable on the y-axis (ordinate) – the vertical axis. 3. Label the axis. Quantities (temperature, length, etc.) and also the proper units (cm, oC, etc.) 4. Choose a range that includes all the results of the data

Graphing Rules 5. Calibrate the axis (all marks equal) 6. Enclose the dot in a circle (point protector) 7.Give the graph a title (telling what it is about) 8. Make the graph large – use the full piece of paper 9. Indent your graph from the left and bottom edges of the page 10. Use a best fit line, do not connect points

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