incorp math lia margolia

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Lia Leon Margolin, Ph.D.Marymount Manhattan College, New York, NY

How to Improve Quantitative and Analytical Skills of Life Science Undergraduates

No human endeavor can be called science if it cannot be demonstrated mathematically.Leonardo da Vinci (1452-1519)

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Quantitative Reasoning Goals for students

Students should be able to:read and analyze dataCreate modelsdraw inferencesSupport conclusions based on sound mathematical reasoning

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Basic Quantitative Skills

Arithmetic -- Fractions, ratios, percentage, scalingUnits -- Conversion, dimensional analysisScientific Notation -- Significant digits, rules of exponentsLinear Equations with one and two unknownsQuadratic equations- factoring, quadratic formulaExponents -- Laws of exponents, relation to logarithmsLogarithms -- Properties of logarithms, relation to exponents Curve Fitting

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Higher Order Skills

Problem solving -- Formulate, Solve, and InterpretModels -- Formulate, analyze, predict, linearity, non-linearityEstimation -- Reasonableness of results, checking answersReading -- Comprehend and analyze mathematical text Writing -- Express quantitative ideas and facts effectively in writing Speaking -- Express quantitative ideas and facts effectively orally Information literacy -- Access and make effective use of quantitative information

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Learning Goal #1 Apply appropriate mathematical model to solve problems

1. Identify alternate quantitative models and select the appropriate model to fit the problemSelect and apply correct model to a specific problemsDraw conclusions from the modelProvide justification for the selected model

2. Construct complete solution to discipline specific problemsSolve problems similar to known problemsGeneralize solutions to types of problems and apply the generalization to other classes of problemsCreate solutions to new and modified problems

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Learning Goal #2 Represent quanttative information verbally, numerically, symbolically,

visually and and draw inferences

1. Share results in appropriate formats: verbally, graphically, symbolicallyPresents results in multiple formatsExplain the linkage between the different formatsDiscusses the various methods of communicating the results

2. Interpret data and judge the whether or not the information is useful in solving discipline specific problemDetermine if a given data set is appropriate for a given problemJustify the choice of data

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Learning Goal #3 Recognize limitations of quantitative models

1. Explain why a particular quantitative model does or doesn't apply to a given data setExamine the model to see if it fits the dataDetermine the limitations of a particular modelCritique the use of a model and suggest alternative model for the appropriate framework

2. Identify underlying quantitative assumptions and challenges the validity of those assumptions within a given contextArticulate the limitations of the modelVerify whether or not the assumptions are met by the modelChallenge the validity of the assumptions, critique or defend the use of the model in the context of the problem

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Math in the Life SciencesMath topics/skill

Basic numeracy/simple arithmetic, order of operations, measured numbers

Sense of the size of numbers; scientific notation; dimensional analysis

Fractions/proportions/ratios/percentages

Systems of two linear equations with two unknowns

Applications

Everything

Converting within the metric system or between systems; deriving molar concentrations from g and ml

Percent Solutions and molar solutions, percent composition of compounds

Calculation of Relative Isotope Abundance, mixture problems

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Math in the Life Sciences (cont.)

Math topics/skill

Finding weighed averagesProperties of exponents and logarithms, quadratic equationsSlope as instantaneous rate of change. Differential and Integral equationsGraphing (data presentation & interpretation)Interpretation of graphical information -- Draw inference

Applications

Average atomic massesAcids, and bases, calculating pHChemical kinetics. differential and integral rate laws

Chemical kinetics identifying the order of the reaction

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Measured & Exact Numbers

Science is based on measurements

All measurements have

1. magnitude2. uncertainty3. units

Math is based on numbers

Exact numbers are obtained by:

1. counting2. definition

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Uncertainty in Measurements

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Uncertainty in MeasurementsThe least count of an instrument is the size of the smallest scale division on the instrument The smaller the least count the more precise the instrument is said to be.

Reference: Morgan. S.(2006). Analytical Chemistry. University of South Carolina

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Significant Figures

—Explain the concept of significant figures—Define rules for deciding the number of significant figures

in a measured quantity.—Explain the concept of an exact number.—Define rules for determining the number of significant

figures in a number calculated as a result of a mathematical operation.

—Explain rules for rounding numbers.—Provide some exercises to test your skill at significant

figures.

Learning Objectives

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Rules for counting significant figuresin a measured quantity

(1) All nonzero digits are significant

(2) Captive Zeroes between nonzero digits 1002 kg has 4 significant figures,3.07 mL has 3 significant figures.

(3) Leading zeros to the left of the first nonzero digits 0.001cm has only 1 significant figure,0.012 g has 2 significant figures.

(4) Trailing zeroes to the right 0.0230 mL has 3 significant figures =2.30 x10^(-2),0.20 g has 2 significant figures3000 km has 1 significant figure

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Rules for counting significant figuresin a measured quantity

If a number is written in exponential notation all figures in the pre-exponential value are significant

Example,50,600 calories can be written as:

5.06 × 10^4 calories (3 significant figures)5.060 × 10^4 calories (4 significant figures), 5.0600 × 10^4 calories (5 significant figures)

Which value has more significant digits?a) 7.630 x10^5 or 0.0261140b) 16,000 or 1600.0

Reference: Zumdahl.S. (2007). Chemistry. Houghton Mifflin Company

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Accuracy vs. Precision

Taylor J. (1999). The Study of Uncertainties in Physical Measurements.University Science Books.

http://books.google.com/books?id=giFQcZub80oC&pg=PA128

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Taylor J. (1999). The Study of Uncertainties in Physical Measurements.University Science Books.

Accuracy & Precision

http://books.google.com/books?id=giFQcZub80oC&pg=PA128

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Ex. 1 A student measures the mass and volume of a piece of Copper and calculate the density of the metal to be 8.37 g/cm3. The student consults a reference table and finds that density ofcopper is 8.92 g/cm3. What would be the student's percent error?

Percent error=-6.1659=-6.17 rounded Usa absolute Value

Measure of Accuracy: Percent error

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Measure of PrecisionThe standard deviation is a statistical measure of the precision for a series of repetitive measurements

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Dimensional Analysis

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Analysis of an air sample reveals that it contains 3.5 x 10-6 g/l of carbon monoxide. Express the concentration of carbon monoxide in lb/ft3.

1 lb = 453.6 g

Conversion from liters to cubic feet using gal

Lgalft1785.348.71 3 ×= =28.32 L

The conversion may be set up in this fashion:

3.5 x 10-6 g/L x 1 lb / 453.6 g x 28.32 L / 1 ft3 = 0.22 x 10-6 lb/ft3

= 2.2 x 10-7 lb/ft3

Unit Conversion: Cubic Units

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Concentration and Molarity

a) Explain how to prepare 25 liters of a 0.10 M BaCl2 solution, starting with solid BaCl2.b) Specify the volume of the solution in (a) needed to get 0.020 mol of BaCl2.

a) molarity (M) = moles solute / liters solution

moles solute = molarity × liters solution;

moles BaCl2 = 0.10 mol/liter x 25 liter=2.5 mol

the weight per mole 1 mol BaCl2 weighs 137 g + 2(35.5 g) = 208 g

2.5 moles of BaCl2 = 2.5 mol × 208 g / 1 mol= 520 g

Part b): Rearrange the equation for molarity to get:

liters of solution = moles solute / molarity = moles BaCl2 / molarity BaCl2 liters solution = 0.020 mol / 0.10 mol/liter=0.20 liter or 200 cm3

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Weighed Average (WA)Average Atomic masses

(WA) is calculated when different data points have difference levels of importance called weights

n

nn

nn

wwwwxwxwxAverageWeighted

wwwweightsandxxxdata

++++++

=.....,...,,...,

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2211

2121

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Average Atomic Masses: Weighed Average

Zumdahl.S. (2007). Chemistry. Houghton Mifflin Company

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Using algebra to find the missing exact mass

(38.9637)(0.9326)+(39.9639)(0.0001)+(X)(0.0673)=39.098336.3415+(X)(0.0673)=39.0983X=40.9618


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Practice Problems: Weighted Averages

An element consists of 1.40% of an isotope with mass 203.973 amu, 24.10% of an isotope with mass 205.9745 amu, 22.10% of an isotope with mass 206.9759 amu, and 52.40% of an isotope with mass 207.9766 amu. Calculate the average atomic mass and identify the element

Calculate the average atomic mass of argon to two decimal places given the following information: argon-36 (25.97 amu, 0.337%), argon-38 (37.96 amu, 0.063%), argon-40 (39.96 amu, 99.600%).


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Solving Systems of Linear Equations with two variables

Relative isotope AbundancesCalculate the percent relative abundanceof the two isotopes of a given element with given average atomic mass of the element and given exact masses of both isotopes

Mixture ProblemsCalculating exact amounts of different components in percent-mixture problems

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The element Indium exists naturally as two isotopes. In113 has a mass of 112.9043amu & In115 has a mass of

114.9041amu. Calculate the percent Relative Abundance of the two isotopes of indium Equation#1: (Atomic mass of In113 )(Relative Abundance of In113 )+ +(Atomic mass of In115 )(Relative Abundance of In115 )= =Average Atomic mass of In Equation#2 The total of 2 isotopic abundances =100%

Calculation of Relative Isotope Abundance

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Relative Isotope Abundance

System of equations to solve112.9043(X)+114.9041(Y)=114.82X+Y=1Substitution Method: Y=1-X112.9043(X)=114.9041(1-X)=114.82X=0.042=4.2% (In 113)Y=0.958=95.8% (In 115)

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Mixture Problem #1A chemist has a solution that is 75% sulfuric acid and a solution that is 25% sulfuric acid. How much of each should she use to obtain 40 milliliters of a solution that is 45% sulfuric acid?Create a system of 2 linear equations

X+Y=400.75X+0.25Y=(0.45)(40)Substitute; Y=40-X0.75X+0.25(40-X)=0.45(40)X=16 mL 75% solutionY=24 Ml 25% solution

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Mixture Problem #2

Mixture Problems. (2002). Retrieved from www.uiuc.edu

32Calculating Percent Composition. (2002). Retrieved from www.uiuc.edu

Percent Composition of Compounds.

33Calculating Percent Composition. (2002). Retrieved from www.uiuc.edu

Percent Composition of Compounds

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Properties of Logarithms: Calculating pH

Acids, Bases, and pH .

Calculate pH and [H+]

When water dissociates it yields a hydrogen ion and a hydroxide.

H2O <--> H+ + OH- ; pH = -log[H+] and [H+] = 10-pH

Calculate pH given [H+] = 1.4 x 10-5 M ;

pH = -log[H+]=- log(1.4 x 10-5)=4.85

Calculate [H+] from a known pH. Find [H+] if pH = 8.5

[H+] = 10-pH [H+] = 10-8.5 [H+] = 3.2 x 10-9 M

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What is the pH of a 0.01 M benzoic acid solution? Given: benzoic acid Ka= 6.5 x 10-5

Benzoic acid dissociates in water as C6H5COOH → H+ + C6H5COO-

Ka = [H+][B-]/[HB]

Benzoic acid dissociates one H+ ion for every C6H5COO- ion, so [H+] = [C6H5COO-].

Let x represent the concentration of H+ that dissociates from HB, then [HB] = C - x where C is the initial concentration.

Ka = x · x / (C -x); Ka = x²/(C - x) (C - x)Ka = x²; x² + Kax - CKa = 0

x = [-Ka + (Ka² + 4CKa)½]/2 where Ka = 6.5 x 10-5 and C = 0.01 M

x = {-6.5 x 10-5 + [(6.5 x 10-5)² + 4(0.01)(6.5 x 10-5)]½}/2 = 7.7 x 10-4

pH = -log[H+] = -log(x)= -log(7.7 x 10-4)= -(-3.11)=3.11

Solving Quadratic EquationsCalculating pH of a Weak Acids

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Analysis of Results

The pH of a 0.01 M benzoic acid solution is 3.11.

In this solution we found the acid only dissociated by 7.7 x 10-4 M. The original concentration was 1 x 10-2 or 770 times stronger than the dissociated ion concentration.

If we substitute C for (C - x) in the Ka equation,

Ka = x²/(C - x) Ka = x²/C ; x² = Ka·C ; x² = (6.5 x 10-5)(0.01)=6.5 x 10-7 x = 8.06 x 10-4

pH = -log[H+] = -log(x)=-log(8.06 x 10-4)=-(-3.09) pH = 3.09

Solving Quadratic EquationsCalculating pH of a Weak Acids

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Slope as Rate of Change: Reaction Kinetics

Reaction Kinetics (n.d.) Retrieved from www.chem.ufl.edu

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Chemical Kinetics: Reaction Rates

Blaunch. D. (2009). Chemical Kinetics. Retrieved from www.chm.davidson.edu

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Pratt, C and Cornely, K. (2004). Essential Biochemistry, Wiley &Sons Inc

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The Units of the Rate Constant

[ ]

[ ]

[ ]

[ ]

( )( )

2 2 2

223 3 3

1

1

sec1 1//

sec2 sec/

sec3 sec/

sec

n

n

n

dAGeneral form of Rate Equation rate k Adt

molrate Lif n then k cA mol L

molrate L Lif n then k molmol LAmolrate L Lif n then k molmol LA

Lfor n th order k has units ofmol

−

−

= − =

= = = =

= = = = ×

= = = =×

−×

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Zero order reaction r = k where k has units of mole/( L) (sec)First order reaction r = k [A] where k, has units of (1/sec)

Second-Order Reaction r = k [A]^2, where k, has units of L/ (mole)(sec)


Differential Rate Laws

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Order in [A]

Rate Law

Integrated Form, y = mx + b

Straight Line Plot

Half-Life t1/2

zero order

(n = 0)

rate = k [A]o=

k

[A]t = - k t +[A]o

[A]t vs. t

(slope = - k)

t1/ 2 =[A]02k

first order

(n = 1)

rate = k [A]1

ln[A]t = - k t +

ln[A]o

ln[A]t vs. t

(slope = - k)

t1/ 2 =ln2k

=0.693

k

second order

(n = 2)

rate = k [A]2

1

[ A]t

= k t + 1

[A]0

1

[A]t

vs. t

(slope = k)

t1/ 2 =1

k[A]0

Integrated Rate Equations

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The first-order rate equation is frequently re-arranged to give

ln[A]t

[A]0

⎛

⎝ ⎜ ⎞

⎠ ⎟ = − k t or ln

[A]0

[A]t

⎛

⎝ ⎜ ⎞

⎠ ⎟ = k t

Using the properties of logarithms,

lnab

⎛ ⎝

⎞ ⎠ = ln(a) − ln(b)

If ln(y) = x then y = ex

[A]t

[A]0

⎛

⎝ ⎜ ⎞

⎠ ⎟ = e − k t or

[A]0

[A]t

⎛

⎝ ⎜ ⎞

⎠ ⎟ = e k t

Other forms of first Order Integrated Rate Equations

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The following data were obtained on the reaction 2 A → B:

Time, s 0 5 10 15 20 [A], mol

L-1 0.100 0.0141 0.0078 0.0053 0.004

(a) Plot the data and determine the order of the reaction.

Time, s [A], mol L-1

ln[A] 1/[A], L mol-1

0 0.1000 -2.303 10 5 0.0141 -4.262 71

10 0.0078 -4.854 128 15 0.0053 -5.240 189 20 0.0040 -5.521 250

Determining the Order of Reaction

Chang. (2002). Chemistry. 7-th edition.

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Reaction 2A B Concentration vs. Time


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Reaction 2A B Graphing ln [A] vs. Time

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Determine the rate constant.Slope=k. k=(250-189)/(20-15)=12 L/(mol)(sec)

Reaction 2A B Graphing 1/ [A] vs. Time

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Conclusion: Important Math ConceptsGraphs

Relationships among variables -- Concept of functionInterpretation of graphical information (linear, polynomials, exponential, logarithmic)Visualization of data -- Histograms, pie charts, scatter plots Log-log and log-linear plotsDraw inference

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Conclusion: Important ConceptsFunctions

Linear -- Slopes and intercepts Polynomial -- Factors and roots Rational -- Fractions of polynomials Power – integral and fractional powers, nth roots Exponential -- growth/decay, relation to logarithm Logarithm -- Natural and base b, growth rate, relation to exponential

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Conclusion: Important ConceptsCollege Algebra & Calculus

Operations with polynomial and rational expressionsSolving linear and quadratic equationsSolving systems of linear equationsApplying rules of exponents and properties of logarithms to manipulate exponential and logarithmic functionsSolve exponential and logarithmic equationsApply concept of slope as instantaneous rate of change in chemical kineticsDifferentiate and integrate polynomial and transcendental functions