University of Washington Department of Chemistry Chemistry 453 Winter Quarter 2012
Homework Assignment 1 Due at 5 p.m. on 1/09/12 in Catalyst Drop-In Box..
This homework is worth a total of 10 points. This homework set is intended as a quick and relatively painless way for you to brush up on your math skills, and for me to understand your math background. Questions of these types will appear in lectures and exams.
QUESTION SET A: The Simple Functions of Applied Math Properties of exponential functions, factorial functions, and logarithms are widely exploited in Chemistry 453. Product and summation symbols are also widely used in chapters on quantum and statistical mechanics.
1A) Using only the following facts, and not the logarithm keys on your calculator, determine ln(60603/2) ~ ? ln(2) ~0.693; ln(3) ~1.099; ln(5) ~1.609; ln(1±x)~±x if x<<1. Show your reasoning.
33 ln() 60603/2 ==×× ln() 6060 ln() 1.01 6 1000 22
333 =++=+++⎣⎦⎡⎤ln1.01 ln 6 ln10[] ln1.01 ln 2 ln 3 3ln10 22 33 =++++=+++[]ln1.01 ln 2 ln 3 3ln 2 3ln 5[] ln1.01 4ln 2 ln 3 3ln 5 22 3 =+⎡⎤0.01 4() 0.693 ++ 1.099 3 () 1.609 = 13.062 2 ⎣⎦
2A) Using only the following facts, and not the logarithm keys on your calculator, determine ln(9901/2) ~ ? ln(2) ~0.693; ln(3) ~1.099; ln(5) ~1.609; ln(1±x)~±x if x<<1. Show your reasoning
11 11 ln() 9901/2 ==×=+ ln() 990 ln() .990 1000 ln() .990 ln() 103 22 22 1 3 0.01 3 =−+ln() 1 0.01 ln() 10 =−+× ln() 2 5 2222 33 3 3 =−0.005 + ln 2 + ln 5 =− 0.005 +() 0.693 +() 1.609 = 3.448 22 2 2 3A) Using only the following facts, and not the exponential key on your calculator, determine e-8010/1000~ ? e3 ~20.086; e5~148.413; ln(1±x)~±x if x<<1. Show your reasoning,
11 e−−8010/1000==ee 8.01−+()8 .01 = e −++ ( 3 5 0.01 ) = eee −−− 3 5 0.01 ≈ × ×−() 1 0.01 20.086 148.413 ==()()()0.0498 0.00674 0.99 0.000332
4A) The factorial function nnnn!12321=−−( )( ) ii increases so rapidly, that evaluating 100! with your calculator is impossible because of memory limitations (try it). There is an expression called Stirling’s Approximation for evaluating factorial n ⎛⎞n expressions: nn!2≈ ⎜⎟ π . Use Stirling’s Approximation to evaluate ln(100!). ⎝⎠e
n ⎡⎤⎛⎞n 1 lnnnnnnn !≈=−++ ln⎢⎥⎜⎟ 2ππ ln() ln 2 ln ⎣⎦⎢⎥⎝⎠e 2 1 ∴ln100!=−++ 100ln100 100() ln 2π ln100 2 1 =−+460.5 100() 1.837 + 4.605 =+= 360.5 3.221 363.7 2
5 5 ⎛⎞1 1 1 1 1 111 6 4 3 13 5A) ∏⎜⎟= /2· /3· /4 6A) ∑ = ++= + + = x=3 ⎝⎠x −1 x=3 x −1 2 3 4 12 12 12 12
QUESTION SET B: Vectors, Matrices, and Determinants Linear algebra is also used in chemistry 453 so familiarity with vectors, matrices, and determinants is advisable. Many physical properties including forces, particle velocities, electric fields and dipole moments are represented mathematically as vectors. Vectors can be multiplied together to obtain a scalar result (i.e. inner or dot product) or a vector result (i.e. cross or outer product). For example, energy is a scalar and the energy of interaction between a dipole moment and a field is obtained by taking an inner product. Torque is an outer product of two vectors which can be evaluated with a determinant. The reorientation of a vector in space is mathematically represented by multiplying a vector with a matrix. Representing a vector both in Cartesian and spherical coordinate systems is also a necessary skill.