Analytical Chemistry Unit 1

A Numerical value is incomplete without…

units and error

Time

second (S)

length

meter (m)

mass

kilogram (kg)

current

amerere (A)

Temperature

Kelvin (K)

Amount of Substance

Mole (mol)

Frequency

Hertz (Hz) - 1/s

Force

Newton (N) - m x kg / s²

Pressure

Pascal (Pa) - kg/(m x s²)

Energy

Joule (J) - m² x kg / s²

Power

Watt (W) - m² x kg/ s³

Charge

Coulumb (C) - s x A

Exa - E

10^18

Peta - P

10^15

Tera - T

10^12

Giga - G

10^9

Mega - M

10^6

Kilo - k

10³

Hecto - h

10²

Deca - da

10^1

Atto - a

10^-18

Femto - f

10^-15

Pico - p

10^-12

Nano - n

10^-9

Micro - u

10^-6

Mili - m

10^-3

Centi - c

10^-2

Deci - d

10^-1

Solution

a homogenous mixture of two or more substances

Solute

minor species in a solution

Solvent

major species in solution

Aqueous solution

solvent is in water

concentration

how much solute is contained in a given volume or mass of solution or solvent

mole

number of atoms in 12g of Carbon 12 = 6.022 × 10 ^ 23 atoms of any kind of particle, mass of a compound/molar mass of a compound

molarity

moles of solute/liters of solution

weight percent

mass of solute / mass of total solution or mixture x 100

volume percent

volume of solute / volume of total solution x 100

parts per million (ppm)

mass of substance / mass of sample x 10^6

parts per billion (ppb)

mass of substance / mass of sample x 10^9

Formality

a substance’s total concentration without regard to its specific chemical form (sum of molarity of ions and undissociated component)

Molarity

concentration of a particular chemical species

• there is no difference between a compound’s molarity and formality if it dissolves without dissociating into ions

Dilution formation

M (conc) x V (conc) = M(dil) x V(dil)

Experimental error

difference between the true value and the measured value of a quantity

Systematic Error

Arises from a flaw in equipment or the design of an experiment

• also called determinate error

• reproducible

• can be detected and corrected

• directional error

• sources

  • instrumental errors: incorrect calibration, drift in detector sensitivity, faulty or dirty glassware

  • method errors: incomplete reactions, side reactions or interferences, extraction or derivatization inefficiencies, nonlinear calibration

  • personal/operational errors: consistent misreading meniscus, poor pipetting technique

  • environmental errors: temperature, humidity variations, CO2 absorption from air changing pH in alkaline solutions

Random Error

arises uncontrolled (or uncontrollable) variables in the experiment

• indeterminate error

• always present

• cannot be corrected

• can be reduced using improved techniques

• non-directional

• sources:

  • instrumental: electronic noise in batteries, fluctuations in power supply

  • human/operator sources: inconsistent technique, judgement calls (color of titration end point)

  • environmental sources: temperature fluctuations, air currents or vibrations, humidity changes

  • sample/process sources: chemical noise, uneven distribution of analyte

blunders

extreme instances of systematic or random error

• gross error

• may have to reject data or redo experiment

Accuracy

how close is the result to the true or accepted value

Precision

How well replicate measurements agree with one another

Repeatability

Describes how well one person can obtain the same results when analyzing the same sample by the same procedure with the same equipment in the same laboratory

Reproducibility

Describes how well different people in different laboratories with different equipment can get the same results when analyzing equivalent samples by the same procedure

Sensitivity

How much the signal changes when analyte concentration changes

Limit of Detection

The smallest amount you can confidently say is present above noise

Limit of quantitation

The lowest concentration you can reliably measure how much (not just “present”)

Limit of linearity

The upper limit that a linear calibration curve will work

Selectivity/Specificity

Extent to which a method can distinguish an analyte from everything else in the sample

Range

The concentration interval over which linearity, accuracy, and precision are all acceptable

Dynamic Range

range of analyte concentration over which a change in concentration gives a change in detector response

Linear Range

Range of analyte concentration over which a change in concentration gives a linear change in detector response

Robustness

Ability of analytical method to remain unaffected by small variations in experimental conditions

Accuracy

how close the result is to the true or accepted value

Precision

How well replicate measurements agree with one another

uncertainty

variability within a set of measurements

absolute uncertainty

margin of uncertainty (with units) associated with a measurement

relative uncertainty

compares the size of the absolute uncertainty with the size of its associated measurement (absolute uncertainty / magnitude of measurement)

percent relative uncertainty

%relative uncertainty = 100 x relative uncertainty

propagation of random error

Identifying significant figures

1. Nonzero digits are always significant

2. Zeros between nonzero digits are significant

3. Leading zeros (before the first nonzero digit) are not significant

4. Trailing zeroes with a decimal point are significant

5. Trailing zeroes without a decimal point are ambiguous

Rules for addition and subtraction with significant figures

1. If numbers are in scientific notation, rewrite them so they have the same exponent

2. Line up the numbers by their decimal points

3. Add or subtract normally

4. identify the number with the fewest digits after the decimal place

5. round the result to that last decimal place

Rules for multiplication and division with significant figures

1. Perform the multiplication/division normally

2. Identify the number with the fewest significant figures - result must have the same number of significant figures as the input with the fewest significant figures

3. round to the correct number of sig figs

Rules for rounding off numbers

1. round only final answer to avoid accumulating rounding off errors. Subscript extra digits during calculation to remind yourself of their insignificance

2. Look at all the digits beyond the last place desired

3. If numbers are more than half way to the next higher digit, round up

4. If numbers are less than halfway to the next higher digit, round less

5. if the number is exactly half way, round to the nearest even digit.

Rules for mixed operations with significant figures

1. Follow the order of operations (pemdas)

2. apply sig fig rules at each stage

   1. for multiplication/division → limit by the number with the fewest sig figs

   2. For addition/subtraction → limit by the least precise decimal place

3. Keep guard digits: don’t round off fully - keep one or two extra digits to reduce rounding error and only round final answer to correct sig figs

Average (mean)

measure of central tendency that represents the typical or central value of a set of data points

• x is sample mean

• u is population mean

Standard deviation

measure of the dispersion or spread of data points around the mean, quantifies how much the individual data points deviate from the average

• higher SD is more variability and lower is less

• s: sample standard deviation

• sigma: population standard deviation

Degrees of freedom

number of values in a statistical calculation that are free to vary (n-1)

variance

square of standard deviation (s²)

relative standard deviation (coefficient of variation)

standard deviation expressed as a percentage of the mean value (100 x (s/mean))

standard deviation of the mean (standard uncertainty or standard error)

quantifies the variability of sample means around the population mean

Null hypothesis

the statement that two sets of data are drawn from populations with the same properties such as standard deviation or mean

confidence interval

a range of values, calculated from sample data, that is likely to contain the true value of the population parameter (like the true mean) with a specified probability (the “confidence level”)

confidence level

the probability (expressed as a percentage) that the confidence interval calculated from your sample data actually contains the true value of the population parameter

• a 95% confidence interval means: If I repeated this experiment many times, about 95% of the calculated intervals would contain the true mean

F-Test

Compares standard deviations of 2 sets: differences are significant if F(calculated) > F(table)

t-test

compares means of two sets: differences are significant if t(calculated) > t(table)

Grubbs test

detects outliers, data point is an outlier if G(calculated) > G(table)

Case 1 t-test

comparing a measured result with a known value

Case 2a t-test

comparing replicate measurements; standard deviations are not significantly different

• use f-test to determine if standard deviations are different or not

Case 2b t-test

comparing replicate measurements; standard deviations are significantly different

• use f-test to determine if standard deviations are different or not

Case 3 t-test

paired t test for comparing individual deifferences; d(i) represents the difference between two results for each sample

calibration curve (standard curve)

a graph showing the value of some property vs concentration of analyte. When the corresponding property of an unknown is measured, its concentration can be determined from the graph.

Standard solutions

a solution whose composition is known by virtue of the way that it was made from a reagent of known purity or by virtue of its reaction with a known quantity of a standard reagent.

Blank solution

solutions containing all reagents and solvents used in the analysis, but not deliberately added analyte. Blanks measure the response of the analytical procedure to impurities of interfering species in the reagents.

standard operating procedures

statement of what steps will be taken and how they will be carried out

range

the concentration interval over which linearity, accuracy, precision, are all acceptable

matrix

everything in the unknown, other than the analyte

matrix effects

interference of the matrix in analytical measurements

How to prepare standard solutions and measure their response

• you have your unknown solution

• you have a blank solution with no unknown molecule

• make standard solutions from 0-150% of concentrations expected for your unknown

Creating a calibration curve

1. subtract average of blank

2. plot corrected response vs [analyte]

3. use least-squares procedure to find the best straight line through the linear portion of the data

Calculate unknown concentration

1. measure unknown and subtract new blank

2. calculate [analyte] from corrected response based on equation of the line

Error in calibration curves

Standard addition

1. add same aliquots of unknown sample in different flasks

2. add increasing volume of standard to different flasks

   1. dilute to same volume with solvent (solvent and standard must have identical matrices)