Lab Math
Learning Objectives:
1. Know abbreviations and be able to use metric and SI units readily, and make
conversions between units
2. Be able to do arithmetic with units
3. Be able to use and interconvert measures of amount in weight (g) and molecules (mol)
4. Understand and be able to use formula weights, including water of hydration
5. Understand the relationship between measures of amount and of concentration,
including specific gravity and density
6. Be able to prepare reagents to concentration specifications
7. Be able to make dilutions appropriately and adjust for their effects
8. Be able to measure concentrations using titrations (including conversions between
moles and equivalents, molarity and normality)
9. Be able to use the Henderson-Hasselbalch equation to calculate pH of buffers and to
determine buffer components needed to achieve a desired pH
Metric system units:
Length
Mass
Time
Temp.
(Volume)
meters (M)
Note: all abbreviations are the same for singular and plural
grams (g)
seconds (s)
degrees (° or °C or °K)
cubic meters (M3)
Système International (SI) units:
Amount
Volume
Time
moles (mol) (1 mole = 6.02 x 1023 particles = atomic/molecular weight in g)
liters (L)
(1 L = 0.001 M3)
seconds (s), minutes (min), hours (h)
Prefixes (apply to both metric and SI measurements)
Prefix
giga
mega
kilo
Abbrev
G
M
k
Factor
109
106
103
deci
centi
milli
micro
nano
pico
femto
d
c
m
μ (or u or mc)
n
p
f
10-1
10-2
10-3
10-6
10-9
10-12
10-15
Example:
How many μg are in 0.5 kg?
What is 15 mg/dL when expressed as g/L?
Math with units
Whenever you do math with measurements, you must do the calculations both with the
numbers and with the units.
Adding and subtracting can only be done when measurements have the same units.
Example:
Mary has a bottle with 5 g of morphine. She weighs out 5 mg to make some controls for
the drug abuse assay. Because morphine is a controlled substance, she must enter both the
amount removed and amount remaining into the controlled substances log book. How much
morphine is remaining in the bottle?
When multiplying or dividing, the units must be multiplied or divided as well.
Example:
A 24-h urine sample has a volume of 1.50 L and a creatinine concentration of 67 mg/dL.
How much creatinine was in the specimen?
Interconverting weight and moles
GIVEN THAT a fixed number of particles of matter of the same type (all one type of atom or
molecule) will have a weight that is proportional to the weight of a single particle;
AND the weight of a single particle is its atomic weight, and the weight of a molecule is the sum
of the weights of the atoms composing the molecules;
THEN, the same number of molecules will be present in any pure substance whose weight in
grams equals the atomic or molecular weight.
The constant number of particles in any pure substance whose weight in grams is equal to its
atomic/molecular weight is Avogadro’s number (experimentally determined to be 6.02 x 1023
particles).
We could express our measurements as the number of molecules in a specimen, but the numbers
would be too large to be handled easily. So we use the more convenient unit of moles, which is
the number of molecules divided by Avogadro’s number and corresponds to a weight in g equal
to the atomic or molecular weight (moles = weight (g) / AW or MW).
Examples:
How many moles of calcium are in 50 g of the pure metal (AW of Ca = 40)?
How many moles of creatinine were in the specimen on the previous page (creatinine =
C4H7N3O; AW of C = 12; H = 1; N = 14; O = 16)? How many mmol?
How many mmol of nitrogen were in the creatinine described above?
Formula Weights
Some substances are highly homogenous, even though they do not have a single type of particle
present. An example is sodium chloride, which has sodium ions and chloride ions present in a
constant 1:1 ratio. It has the formula NaCl, although the sodium and chloride are not covalently
attached. We can handle substances of constant composition as if they were atoms or molecules,
using the formula weight. A solution containing one formula weight per liter is often referred to
as having a concentration of 1 mol/L. It may contain 1 mole of each type of particle in the
formula, or it may contain 2 moles or more of some of the particles, depending on the formula.
Example:
How many moles of sodium ions are in 117 g of NaCl (AW Na 23; Cl, 35.5)?
How many moles of sodium ions are in 142 g of sodium sulfate, Na2SO4 (AW Na 23; S, 32; O,
16)?
Some homogenous substances may include water of crystallization in their formula.
Example:
How much sodium sulfate decahydrate Na2SO4·10H2O would you have to weigh out to have 1
mmol of sodium?
Formula weights are also important when working with salts of organic molecules.
Examples:
How much morphine sulfate ([morphine]2·H2SO4)would you have to weigh out to have 5 mg of
morphine (FW morphine 285; H2SO4, 98)?
How much sodium phenytoin (an antiseizure medication) would you need to make 0.5 L of 10
μg/mL phenytoin (MW phenytoin 252; AW Na 23)?
Measuring Amounts
The most direct measure of amount is weight. We can also measure amounts by measuring the
volume. This is often more convenient than weight for liquids and is absolutely necessary for
gasses. To convert volumes of liquid to moles, we must first convert to weight using the specific
gravity or density. (To convert the volume of gas to moles, we must know the temperature and
pressure and use the universal gas law, PV = nRT. This will be covered further in the blood gas
lectures).
Example:
How many grams of ethyl alcohol are in 100 mL (the specific gravity (sp. gr.) of 100% ethanol is
0.79; the density (d) is 0.79 g/mL)?
How many mL of 95% (v/v) ethanol (sp. gr. 0.81) will you need to make 100 mL of a 100 mg/dL
solution? (Warning!—this could be a trick question! See the next section)
Percentages
Sometimes we express concentrations as percentages. The tricky part is that there are three
kinds:
% (v/v) the number of mL of a substance in 100 mL of solution
% (w/v) the number of g of a substance in 100 mL of solution
% (w/w) the number of g of a substance in 100 g of solution
Be careful which kind is being used. If not specified, usually w/v can be assumed (but not
always).
Knowledge of percentages is needed when preparing reagent solutions from commercial reagents
that are not 100% pure.
Example:
Bob wants to prepare 1L of a 1 mol/L solution of hydrochloric acid. He has a bottle of
commercial concentrated hydrochloric acid, which is described on the label as being 37% (w/w)
hydrochloric acid and as having a specific gravity of 1.20. How many mL of concentrated
hydrochloric acid will he need (FW HCl = 36.5)?
Concentrations
The physiological effect of most substances we measure in the clinical laboratory is determined
the concentration; that is, the amount in a given volume of solution. Common units for
concentration measurements include mol/L, mmol/L, mg/L, mg/dL, and μg/mL. Others are also
used. You need to be able to convert between these measurements readily.
(Note that concentrations in mol/L are also called molar, and abbreviated M. A 2.5 mM solution,
contains 2.5 mmol/L)
Examples:
A serum specimen contains 15 μg/mL of phenytoin. What is its concentration in mg/L?
A serum specimen contains 150 μg/mL of salicylate (the active metabolite of aspirin). What is
its concentration in mg/dL?
The reference range for magnesium is 1.8 – 2.4 mg/dL. What is the reference range expressed
as mmol/L (AW Mg = 24)?
In some assays, the signal we measure is determined by the concentration of the analyte
(substance being measured). An example is the voltage produced by the electrodes in a blood
gas machine. In this case, the volume we use is not important, as long as there is enough to fully
cover the electrode.
In most assays, however, the signal is determined by the number of molecules (amount) in the
reaction mixture. The concentration is determined by assuming that we will always add the same
volume of the patient’s specimen to the mixture. If we add the wrong volume, we will get a
wrong answer!
Example:
In a sandwich immunoassay, the analyte is the peanut butter that makes the two halves
of the sandwich stick together. The number of sandwiches formed is equal to the number of
analyte molecules. The amount of signal is proportional to the number of sandwiches. If the
technologists adds 15 uL of specimen, rather than the 10 ul called for, what will be the effect on
the answer?
Calibration curves
How do we know the relationship between the signal generated in an assay, and the
concentration of the analyte?
We measure the signals generated by one or more calibration samples with known
concentrations.
Examples:
In the calibration curve below, what is the relationship between the signal and the concentration
over the concentration range from 0.0 to 0.05 g/dL? What is the concentration in a patient
specimen giving an A455 reading of 1.5?
What is the relationship between the signal and the concentration over the concentration range
from 0.05 to 0.10 g/dL? What is the concentration in a patient specimen giving an A455 reading
of 3.8? What is the apparent concentration, assuming a linear calibration curve, in a patient
specimen giving an A455 reading of 3.8?
5
O
4
O
3
O
2
O
O
1
O
0
0
0.02
0.04
0.06
0.08
concentration, g/dL
0.1
Dilutions
Dilutions can allow us to make accurate measurements on specimens with concentrations that
exceed the linear range of the calibration curve, by creating a specimen with a concentration
within the linear range
Examples:
What is the absorbance of a specimen with a concentration of 0.1 g/dL using the
calibration curve above? What would its apparent concentration be in the assay?
If the specimen were diluted with an equal volume of water, what would the new concentration
be? What would the A455 of the diluted solution be? What would its apparent concentration be,
assuming a linear calibration curve? What would the apparent concentration of the original
specimen be after correcting for the dilution?
An assay for phenobarbital gives a result of 52 mg/L, greater than the assay working range of
2 – 40 mg/L. It is repeated after 2-fold and 4-fold dilutions, which give results of 30 mg/L and 15
mg/L. Is the true value 15 mg/L, 30 mg/L, 52 mg/L or something else?
Dilutions allow accurate preparations of very low concentration solutions.
Example:
How could you prepare a solution containing 1 ng/mL of the cardiac drug digoxin,
accurate to within 2%? Assume you have a scale that can weigh material to an accuracy of
0.1 mg, and volumetric glassware with an accuracy of 0.2%, provided measured volumes are
10 mL or greater. Your largest volumetric flask is 2 L.
Dilutions allow easy concentration adjustments. To do this, we use a rearrangement of of the
definition of concentration, Conc. = Amount/Vol., to one of Amount = Conc. x Vol. Since the
amount of analyte is not changed by adding a diluent, the following equation must be true:
C1V1 = C2V2
where C1 and V1 are the concentration and volume before dilution
C2 and V2 are the concentration and volume after dilution
Remember this equation. It is extremely useful.
Examples:
Sam wants to prepare a 250 mL of a solution containing 150 mmol/L of sodium. He has
a stock solution of 10%(w/v) sodium chloride. How much of the stock solution should he put into
his 250 mL volumetric flask before filling with purified water (AW Na 23; Cl, 35.5)?
Using concentrated hydrochloric acid, Mary prepared a solution of 100 mL to be 1.00 mol/L.
However, when she titrated it against a primary standard of 1.00 mol/L sodium hydroxide, she
needed 10.35 mL of NaOH to neutralize 10.00 mL of her HCl. How much water should she add
to the remaining 90 mL of acid to make it exactly 1.00 mol/L?
Linear dilutions allow efficient preparation of a set of calibrators
Example:
Suggest a method for preparing the following calibrators for an ethanol assay using a
linear dilution scheme: 100 mg/dL, 80 mg/dL, 60 mg/dL, 40 mg/dL and 20 mg/dL. Hint: Begin
by making a stock solution containing 100 mg/dL.
Serial dilutions can be used to rapidly explore a wide range of concentrations. To make serial 2fold dilutions, mix 1 mL of material with 1 mL of diluent. Take 1 mL of this 2-fold dilution and
add 1 mL of diluent to make a 4-fold dilution. Take 1 mL of the fourfold dilution and add 1 mL
of diluent to make an 8-fold dilution. And so on.
Example:
A qualitative colorimetric test for serum ketones is used to detect acetoacetate produced
during diabetic ketoacidosis. The test was positive on the original specimen, and on a series of
serial dilutions up to a 16-fold dilution. There was no color development on the 32-fold dilution.
The manufacturer reports that 20 mg/dL of acetoacetate gives a trace positive result. What was
the approximate concentration of acetoacetate in the specimen? What is our range of
uncertainty?
Dilutions are very useful, but are also a common source of laboratory errors.
Examples:
An specimen assayed for phenobarbital gives a result of 30 mg/L. The working range is
2 – 40 mg/L.
What is the expected result after a 1:1 dilution?
What is the expected result after a 1:2 dilution?
What is the expected result after a 2-fold dilution?
What is the expected result if 0.5 mL of the specimen is mixed with 0.5 mL of water?
Titrations
Titrations allow measurement of concentrations by neutralizing the analyte with a known
solution. Note that as the titrant neutralizes the unknown, the unknown also neutralizes the
titrant. Titration is most widely used to measure a base by titrating with an acid or vice versa.
When all of the unknown has been neutralized, an endpoint signal is generated (disappearance of
colored unknown, or appearance of the color of the titrant, or change in color of an indicator).
Examples:
Ten mL of a solution of acetic acid is titrated with 1 mol/L standard solution of sodium
hydroxide in the presence of phenolphthalein indicator. The solution turns from colorless to
slightly purple after 5.0 mL of sodium hydroxide is added. What is the concentration of the
acetic acid solution?
Silver ions are generated at a silver electrode by removing electrons. These Ag+ ions
react with chloride ions (Cl– ) in the specimen to make insoluble AgCl, which precipitates. When
all of the Cl– ions have been removed, the excess Ag+ ions remain in solution and increase the
conductivity, giving an endpoint. One hundred µL of specimen required removal of 6 x 1018
electrons (10 µmol of electrons) to cause a conductivity rise. What was the chloride
concentration in the specimen?
Equivalents and Normality
Some acids and bases can donate or accept more than one proton per mole (H3PO4, or Na2CO3,
for example). There is no longer a mole for mole relationship in titrations. To re-establish a 1:1
relationship in titrations, the concepts of equivalents and normality were developed. An
Equivalent (Eq) is the amount that donates or accepts one mole of protons. Normality (N) is the
concentration in Eq/L. To convert mol to Eq, or M (mol/L) to N (Eq/L), multiply by the
number of protons that can be donated or accepted, or by the number of charges on the
ion.
Examples:
What is the normality of 0.5 M H2SO4? Of 0.5 M H3PO4? Of 0.5M Al(NO3)3?
How much of the anticoagulant potassium oxalate, K2(C2O4) = FW 166, would you weigh
out to make 200 mL of an 0.1 N solution of K2(C2O4)?
Henderson-Hasselbalch Equation
The Henderson-Hasselbalch Equation can be used to calculate the pH of a buffer based
on the concentrations of the acid and base forms of the buffer. It is based on the fact that
equilibrium, the rates of acid dissociation and acid formation are equal:
HA  H+ + A–
k1 [HA] = k2 [H+][A–]
[H+] = (k1/ k2 )[HA]/[A–]
– log[H+] = – log(k1/ k2 ) – log[HA]/[A–] = – log Ka – log[HA]/[A–], where Ka = (k1/ k2 )
pH = pKa + log [A–]/[HA]
Henderson-Hasselbalch Eq.
To use this equation, you need to know the pKa. For common buffers, this is readily found in a
variety of references. The most important pKa values for the clinical lab are:
acetic acid/acetate (CH3COOH  H+ + CH3COO–), pK = 4.76
carbonic acid/bicarbonate (H2CO3  H+ + HCO3–), pK = 6.1
dihydrogen phosphate/ hydrogen phosphate (H2PO4=  H+ + HPO4–), pK = 6.8
Examples:
What is the pH of a serum specimen with a bicarbonate concentration of 24
mmol/Land a carbonic acid concentration of 1.2 mmol/L? Suppose the bicarbonate
concentration is 6 mmol/L and the carbonic acid concentration is 1.2 mmol/L, what is
the pH?
You want to make 2 L of a 0.2 M sodium phosphate buffer with a pH of 7.0. How
much NaH2PO4 and how much Na2HPO4 would you need to weigh out (Na = 23; H = 1;
P = 32; O = 16)?