1. Chapter 4: Calculations Used in Analytical Chemistry

2. 4A Some important units of measurement 4A-1. SI Units
SI units system: the acronym for the French Système International d’Unités (The International System of Units).
SI units is based on 7 fundamental base units.
Derived SI units: the units derived from 7 SI base units such as volts, hertz, coulombs, and joules etc.
Accepted SI units: The SI system also allows the use of some common units that are related to but not directly derived from the fundamental SI units, called accepted SI units.
To express small or large measured quantities in terms of a few simple digits, prefixes are used with these base units, derived units and accepted units.
Derived: 衍生的

3. SI base units - 1 Physical Quantity Name of Unit Symbol
Definition of Unit
Length
meter m
A meter is defined as the distance traveled by light in a vacuum in 1/299,792,458 of a second.
Mass
kilogram kg
A kilogram is defined as the mass of a platinum–iridium cylinder that is kept as the international standard for the kilogram at the International Bureau of Weights and Measures in Sèvres, France.
Time
second s
A second is defined as the amount of time equal to 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of cesium-133.
Amount of
substance
mole
mol
A mole is defined as the number of individual entities of a substance that is equal to the number of carbon atoms in kg of carbon-12.

4. SI base units - 2 Physical Quantity Name of Unit Symbol
Definition of Unit
Temperature
kelvin K
The kelvin temperature scale assigns the lowest possible temperature (absolute zero) a value of 0 K, and the triple point of water (where the gas, solid, and liquid forms of water all exist in equilibrium) a value of K.
Electric
current
ampere A
One ampere is defined as the constant current that produces a force of 2 x 10-7 newton per meter of length when maintained in two straight parallel conductors of infinite length and negligible circular cross section that are placed one meter apart in a vacuum.
Luminous
intensity
candela cd
A candela is the luminous intensity measured in a given direction from a source that emits monochromatic radiation with a frequency of hertz, and that has a radiant intensity of 1/638 watt per steradian in the observed direction.

5. Derived and Accepted SI Units - 1
Measured Quantity
Base Unit (Symbol)
Relationship to Other SI Units
Frequency
hertz (Hz)
1 Hz = 1/s
Force
newton (N)
1 N = 1 m · kg/s2
Pressure
pascal (Pa)
1 Pa = 1 N/m
Energy
joule (J)
1 J = 1 N · m
electron-volt (eV)
1 eV = x J
Electric charge
coulomb (C)
1 C = 1 A · s
Power
watt (W)
1 W = 1 J/s
Electric potential
volt (V)
1 V = 1 W/A = 1 J/(A · s)
Electric resistance
ohm (Ω)
1 A = 1 V/A
Temperature
degree Celsius (°C)
°C = K –

6. Derived and Accepted SI Units - 2
Measured Quantity
Base Unit (Symbol)
Relationship to Other SI Units
Time
minute (min)
1 min = 60 s
hour (h)
1 h = 3,600 s
day (d)
1 d = 86,400 s
Volume
liter (L)
1 L = 10-3 m3
Mass*
unified atomic mass unit (u)
1 u = * kg
*In many biologically related fields (and particularly in work with proteins), the “dalton” (symbol, Da) is commonly used in place of the unified atomic mass unit, where 1 Da = 1 u.
*The ångstrom (Å) is a non-SI unit of length widely used to express the wavelength of very short radiation such as X-rays (1 Å = 0.1 nm). Typical X-radiation lies in the range of 0.1 to 10 Å.
Unified atomic mass unit: 統一原子質量單位

7. Prefixes Used in the SI System
Prefix Name (symbol)
Multiplier
yotta- (Y)
1024 (1 septillion)
deci- (d)
10–1 (1 tenth)
zetta- (Z)
1021 (1 sextillion)
centi- (c)
10–2 (1 hundredth)
exa- (E)
1018 (1 quintillion)
milli- (m)
10–3 (1 thousandth)
peta- (P)
1015 (1 quadrillion)
micro- (μ)
10–6 (1 millionth)
tera- (T)
1012 (1 trillion)
nano- (n)
10–9 (1 billionth)
giga- (G)
109 (1 billion)
pico- (p)
10–12 (1 trillionth)
mega- (M)
106 (1 million)
femto- (f)
10–15 (1 quadrillionth)
kilo- (k)
103 (1 thousand)
atto- (a)
10–18 (1 quintillionth)
hecto- (h)
102 (1 hundred)
zepto- (z)
10–21 (1 sextillionth)
deka- (da)
101 (ten)
yocto- (y)
10–24 (1 septillionth)
*Metric units of kilograms (kg), grams (g), milligrams (mg), or micrograms (µg) are used in the SI system.
*Volumes of liquids are measured in units of liters (L), milliliters (mL), microliters (µL), and sometimes nanoliters (nL).

8. Converting Units
Factor-label method (dimensional analysis, picket fence method):
Write the number and unit in a calculation
Conversion factor should have the same unit in its denominator as the unit in the previous numerator.
Example 
Dimensional analysis: 因次分析
Picket fence method: 樁柵法
Denominator: 分母
Numerator: 分子

9. Example: Prefixes and DNA Analysis
…Ideally, the amount of DNA will double each time PCR is carried out on a sample.
Thus, if you start with only one molecule of DNA (where 1 molecule = 1.66 x mol) you get up to 215 molecules of DNA after 15 cycles of PCR or 230 molecules after 30 cycles.
Using the prefixes in the SI system, determine how many moles of DNA would be present after 15 and 30 PCR cycles if you begin with a single copy of DNA.

10. Example: (continuous)
SOLUTION:
After 15 PCR cycles the amount of DNA can increase by as much as 215 (or 32,768-fold).
mol duplicated DNA = (32,768)x (1.66 x mol)
= 5.44 x mol = 54.4 zmol or amol
After 30 PCR cycles the amount of DNA can increase by as much as 230 (or 1,073,741,824-fold).
mol duplicated DNA = (1,073,741,824)x (1.66 x mol)
= 1.78 x mol = 1.78 fmol  

11. 4A-2. The Distinction Between Mass and Weight
Mass (m): An invariant measure of the quantity of matter in an object.
Weight (w): The force of gravity on the object
w = mg g: acceleration due to gravity (9.8 m/s2).
A balance is used to compare the mass of an object with the mass of one or more standard masses.
weighing: 秤重
weights: 重(質)量、砝碼

12. 4A-3. The Mole (abbreviated mol)
The same number of particles as the number of carbon atoms in exactly 12 grams of 12C. The number is Avogadro’s number NA= x 1023.
Molar mass (M): for the mass in grams of 1 mole of that substance, (g/mol)
1 u = (1/6.022 x 1023) g
one 12C atom, exactly 12 u
molar mass of one mole 12C atoms, 12 g/mol
Molar mass: molecular weight (MW), formula weight (FW), atomic weight (AW).
4A-4. The Millimole (abbreviated mmol)
1 mmol = 10-3 mol

13. 4A-5. Calculating the Amount of a Substance in Moles or Millimoles
Example: The molar mass of glucose (C6H12O6)

14. 

15. 4B Solutions and their concentrations 4B-1. Concentration of Solutions
Molarity (M, Molar concentration)
Molar analytical concentration (cX): the number of moles of a solute in 1 L of solution regardless what might happen to the solute during the solution process.
Molar equilibrium concentration ([X]): the molar concentration of a particular species in a solution at equilibrium.
The molar analytical concentration describes how a solution of a given concentration can be prepared.
To specify the molar equilibrium concentration of a species, it is necessary to know how the solute behaves when it is dissolved in a solvent.

16.     * 

17.  

18. 補充材料 Molarity (M) Formality (F) Normality (N) molality (m) Temperature
dependent
Temperature
dependent
Temperature
dependent
Formality also called analytical concentration or formal concentration, used for electrolyte.
Detailed for Normality listed in Appendix 7, pages A-19~26
Temperature
Independent

19. Percent Concentration
w/w: temperature independent
v/v and w/v: temperature dependent

20. Percent Concentration
weight percent (w/w) is often used to express the concentration of commercial aqueous reagents. EX. 70% (w/w) nitric acid meaning 70 g HNO3 per 100 g of solution
volume percent (v/v) is commonly used to specify the concentration of a solution prepared by diluting a pure liquid compound with another liquid. EX. 5% (v/v) methanol meaning diluting 5.0 mL pure MeOH with water to 100 mL.
weight/volume percent (w/v) is often used to indicate the composition of dilute aqueous solutions of solid reagents. EX. 5% (w/v) silver nitrate meaning dissolving 5.0 g AgNO3 with water to give 100 mL solution.
* 50% (w/w) NaOH corresponds to 76.3% (w/v) NaOH

21. Parts per million, parts per billion and part per trillion
Unit name
abbre.
Sample type
Solid
(w/w)
Liquid
(w/v)
gas
(v/v)
part per million
(106)
ppm
mg/kg
μg/g
mg/L
μg/mL
μL/L
nL/mL
part per billion
(109)
ppb
μg/kg
ng/g
μg/L
ng/mL
nL/L
pL/mL
part per trillion
(1012)
ppt
ng/kg
pg/g
ng/L
pg/mL
pL/L
fL/mL
psu (practical salinity unit): the unit for salinity
ppt (part per thousand): the unit for salinity
*In oceanography, part per thousand (‰) is also used.

22. 

23. Solution-Diluent Volume Ratios
The composition of a dilute solution is sometimes specified in terms of the volume of a more concentrated solution and the volume of solvent used in diluting it. Ex. usually 1:4 HCl solution meaning 4 volumes of water and 1 volume of concentrated hydrochloric acid.
p-Functions
The negative logarithm (to the base 10) of the molar concentration of that species. Thus, for the species X, pX = - log [X]

24. 
10^-6.372=10^0.628x10^ digits for significant figures 

25. Density and Specific Gravity of Solutions
Density: the mass of a substance per unit volume, in units of kg/L or g/mL.
Specific gravity: the ratio of the mass of a substance to the mass of an equal volume of water, dimensionless.

26. 

27. Dilution
The process of preparing a less concentrated solution from a more concentrated solution (EX. stock solution).
cconcd: concentration of the concentrated solution
Vconcd: volume of the concentrated solution
cdil: concentration of the dilute solution
Vdil: volume of the dilute solution

28.   

29. 4C Chemical stoichiometry
Stoichiometry: the quantitative relationship among the amounts of reacting chemical species.
Figure 4-2 Flow diagram for making stoichiometric calculations.

30.  

33. Homework (Due 2013/10/1) End of Chapter 04
Define solute, solvent, solution and concentration.
Define molarity, molality and normality.
Define empirical formulas, molecular formulas and structural formula.
Define molar analytical concentration and molar equilibrium concentration.
End of Chapter 04