Proportional Relationships Chapter 14

Intro to ratios and proportions Ratio is a relationship between two quantities Proportions are statements that two ratios are equal Ratio = 30 mi/1 gal or 1 gal/30 mi Proportion 1 cake/2 oz choc = 2 cake/4 oz Proportional statements are equations due to the = sign

Problem solving A Proportions How many oz of chocolate are needed to make 5 cakes if 1 cake uses 2 oz? 2 oz choc = X 1 cake 5 cakes To solve for X cross multiply and divide Or 2 oz choc x 5 cakes = 1 cake x X Or X = (2 oz x 5 cakes)/ 1 cake X = 10oz

Problem solving 2 Dimensional Analysis 1 cake = 2 oz of choc So 1 cake/2 oz of choc or 2 oz of choc/cake We want to determine chocolate in 5 cakes We want to eliminate cakes 2 oz of choc 5 cake = 10 oz of choc 1 cake

Keeping track of units In proportions units of the denominator and numerator must be the same The dimensional analysis, the denominator and numerator will be different In both cases, unneeded units will cancel and needed units are left. If this is not the case, the problem is wrong

Basics of Dimensional Analysis Makes use of the cancellation properties of fractions Both units and numbers may be cancelled Can be used with anything that has an equivalence statement. The conversion factors are derived from equivalence statements.

Cancellation of Fractions Numbers may be cancelled in the numerator and denominator if they are the same or multiples of one another Ex: 3 x 4 = 4 3 Ex: 3 x 4(1)= 3 x 1 = 1.5 8(2) 2

Cancellation of Units Cancellation of units allows conversion of one unit to another 12 in x 3 ft 36 in 1 ft 120 g x 4 cups = 480 g 1 cup

Equivalence Statements Set one quantity equal to another. Ex: 12 in = 1 ft 3 ft = 1 yd 10 mm = 1 cm These statements will always have an equal sign in them.

Conversions Factors Convert one unit to another Are formed from the equivalence statement by dividing each side by the other There are two conversion factors for each equivalence statement. Use the conversion factor that will give you the desired unit and cancel the other units.

Deriving Conversion Factors Equivalence statement: 12 in = 1 ft Conversion factors: 12 in/1 ft or 1 ft/12 in Equivalence statement: 1 cup = 120 g Conversion factors: 1cup/120g or 120g/1cup

Percents Percents are familiar ratios The % sign symbolizes a fraction and the word percent means “of every hundred” I like to say a % is parts per hundred % is the number of units studied X 100 Total units

Rules for percent % to decimal Decimal to % % to fraction Move decimal two places to the left Decimal to % Move the decimal two places to the right % to fraction Write % as fraction with denominator of 100 Fraction to % Divide numerator by denominator and multiply by 100 and add % sign

Continued Calculating the % of a number What is 25% of 200? Convert the percent to a decimal Multiply the decimal times the number of interest What is 25% of 200? 25% = 0.25 .25 x 200 = 50

Percents to Laboratory Solutions How would you prepare 300 ml of a 25% ethanol solution? Ethanol content = .25 x 300 ml = 75 ml Water content = total – ethanol Water content = 300 ml – 75 ml = 225 ml To prepare the solution, mix 225 ml of water and 75 ml of ethanol

Density Density, d, is the ratio between the mass and volume of a material Density = mass/volume May express in various units Density changes with temperature due to expansion of the material Benzene d20o = 0.880 g/ml Density is 0.880 g/ml at 20oC

Continued Densities are often compared to water The density of water is 1.00 g/ml Immiscible liquids of different densities will form layers Density = mass/volume Mass = density x volume Volume = mass/density

Concentration and Dilution Concentration is the amount of a particular substance in a stated volume of a solution or mixture Solute is the substance that is dissolved or dissociated Solvent is the substance that dissolves the solute Amount is how much of a substance is present

Practice How do you make a 400 ml solution that has a concentration of 10 g of KBr in 100 ml of total solution? Amount of solute: 10 g of KBr 400 g of sol. = 40 g 100 g of solution Dissolve KBr in 100 g H2O and fill contain.

Dilutions Dilution is when one substance is added to another to reduce the concentration of the first The original solution is called the stock solution Many ways to define solutions Need to be sure you are talking about the total of the final mixture or amount of diluent (substance that dilutes)

Dilution Terminology 1 part of food coloring combined with 9 parts of water means the food coloring is 1 part in 10 ml of total volume or 1/10 food coloring An undiluted substance is called 1/1 When talking about dilutions, the symbol(:) means parts. 1 ml of food color + 9 ml of water (1:9) This is not a 1:10 dilution

Examples A 1:2 dilution means there are three total parts of volume ½ dilution means that there are two total parts of volume ½ = 1:1 1:2 = 1/3 1:3:5 means 1 part, 3 parts b, and 5 parts c

Dilutions and proportional relationships Dilution and proportion are related 1 ml/10 ml = 10 ml/100 = 1/10 100 ml of enzyme are added to 900 ml of buffer. What is the dilution? 30 ml of protein are placed in 400 ml of hepron. What is the dilution?

Calculations for Preparing one dilution How do you prepare 10 ml of a 1/10 dilution of food coloring? What does 1/10 mean? How much diluent is needed? How much food coloring is needed?

Other calculations 100 ml of a 1/5 dilution of food coloring? 1 = x = 20 5 100 100 Add 20 ml of food color and 80 of water

Dilution and Concentration Concentration of diluted tubes is important RULE: The concentration of a diluted solution is determined by multiplying the concentration of the original solution times the dilution expressed as a fraction.

Examples A solution containing 5 mg of enzyme/ ml of solution is diluted 1/10. What is the concentration of the diluted solution? 5 mg 1 = .5 mg/ml 10

Dilution Series Dilution series is a group of solutions that have the same components but at different concentrations. Accomplished by making each diluted solution independently of the others starting with a stock solution. Serial dilutions are similar to a dilution series but the dilution factor is consistent from step to step

Independent dilution Make a 1/10, 1/50 and 1/100 dilution of food coloring from an original bottle. 1/10 dilute 1 ml to 10 total ml of volume 1/50 dilution 10 ml 1 = 0.2 ml of fc in 10 ml of vol 50 1/100 dilution 10 ml 1 = 0.1 ml in 10 ml of vol 100

Non independent dilutions The concentration of a diluted solution in the final tube is determined by multiplying the concentration of the original solution times the dilution in the first tube, times the dilution in the second tube and so on. A 1 ml sample of enzyme was diluted 1/10 by the producer. The end user diluted this mixture 1/5. If the original material was 10 mg/ml, what is the concentration of the final dilution

Continued Describe the dilution of a solution containing 15 g of HCl/liter to .15mg/l. We need 2 ml of the final solution. Are there more than one way to do this dilution?

Serial Dilution A serial dilution is a series of dilutions that all have the same dilution factor. (all are 1/10 or ½ dilutions) Could the previous problem be done as a serial dilution?