Chap 1 Problem Solving and Numerical Mathematics

Chap 1 Problem Solving and Numerical Mathematics

Topics in Physical Chemistry

• Quantum Chemistrylinear algebra, calculus, differential equation

• Spectroscopy group theory, Fourier transform

• Thermodynamics multivariable calculus

• Chemical Kinetics linear algebra, differential equation

• Statistical Thermodynamics probability, statistics

Define a Problem in Physical Chemistry

• Word problems state • Physical systems involved

• State of systems

• The desired outcome.

e.g. Hydrogen atom emission spectrum

Physical systems: _____

State of systems: _____

Desired outcome:

Q: Is there any rule governing these lines?

A: 1885 J. Balmer proposed: 𝜆 = 𝐵𝑛2

𝑛2 − 22 , 𝑛 > 2

Problem Solving

• Procedure of problem solving1. Know what is to be calculated

2. Determine whether additional information not stated in the problem is needed

3. Find if there is additional information that help problem solving

4. Determine what procedures and calculations will get you to the desired result

5. Carry out the procedures and calculations

6. Make sure the result is reasonable

7. Round off any insignificant digits

Measurements: Numbers and Units

• A measurable quantity includes a number and a unit

length: 160 cm = 1.60 m = 3 ft 3.94 in

weight: 50 kg = 50000 g = 110.1 lb

The size of a hydrogen atom ~ 1 ___ (unit)

The weight of an oxygen atom ~ ____ kg

The weight of a 100 ml beaker ~ 100 ___

Types of Numbers• Real (scalar) numbers consist of a magnitude and a sign ±

• Rational numbers:

Integers:0, 1, 2, 3, …

−1, −2, −3, …(whole number)

Fractions: quotients of two integers:1

16,

2

3,

5

7,

37

53, …

decimal form: 0.625, 0.66666…, 0.714285714285…

finite numbers or repeating pattern of digits

• Irrational numbers:Algebraic irrational numbers: nth roots of rational numbers

2,3

3, …

Transcendental irrational numbers:

= 3.1415926535…,

e = 2.7182818284…

infinite number and no repeating pattern of digits

Units of Measurement

• SI (Systéme International d’Unités) system

or MKS (meter, kilogram, sec)

7 Base Units

Derived SI Units of Measurement

Prefixes for Multiple and Submultiple of Unit

Common Non-SI Units

• Pressure: atm (atmosphere), Torr

1 atm = 760 Torr = 101,325 N/m2 (Pascal)

• Volume: l (liter)

1 l = 0.001 m3

• Temperature: ℃

0 ℃ = 273.15 K

Using Consistent Units in Calculation

When making numerical calculation, make sure that you use consistent units for all quantities

e.g. Calculate the energy of a photon with wavelength λ = 532 nm.

𝐸 = ℎ𝜈 = ℎ𝑐

𝜆h: Planck constant (6.626⨉10-34 Js-1)

c: speed of light (2.998 ⨉108 ms-1)

λ: wavelength

ν: frequency of light

E = ____________

Conversion of Units

• A conversion factor: a fraction with the numerator and denominator equal to the same quantity expressed in different units

0.35 hr = 0.35 hr ×60 min

1 hr= 21 min

11

cm= 1

1

cm×

100 cm

1 m= 100

1

m

Accuracy and Significant Digits of Measurement

e.g. Measure length with a ruler

25.33 0.03 cm 25.3 cm ( 3 significant digits)

0.003045 4 significant digits

76,000 2 significant digits

• When doing numerical calculation, carry at least one insignificant digit in your intermediate calculation to avoid accumulation of errors.

• When report the final answer, only show the significant digits.

accuracy

Scientific Notation

• a × 10m

1⩽ |a| < 10

m: integer

i.e. 0.005980 = 5.980⨯10−3

7,342,000 = 7.342⨯106

Rounding

• To remove insignificant digits.Round up when the insignificant number ≥ 6

Round down when the insignificant number ≤ 4

• When the insignificant digit is 5• Round to the even digit

e.g. 78955 round to four significant digits → 78960

78945 → 78940

Significant Digits in a Calculated Quantity

• Multiplication and DivisionThe product will have the same number of significant digits as the factor with the fewest significant digits.

e.g. V = (7.78 m)(3.486 m)(1.367 m) = 37.07451636 m3

37.1 m3

• Addition and SubtractionA digit that is significant must arise from a significant digit in every term of the sum or difference.

e.g. 0.788 m + 17.3184 m = 18.1064 m 18.106 m