CHM 3411 – Problem Set 1

Due date: Friday, January 16 (in class or by 5 pm in my office, in my mailbox or in the holder outside my office).

Do all of the following problems. Show your work.

“You might say that our minds aren't perfectly adapted to understand quantum mechanics. Evolutionists would argue that we don't need to understand it to survive. But you do need it to pass…” Dr. John Hermanson

1) For a conservative one dimensional system the differential equation for the motion for a particle may be written as follows

d²x/dt² = - (1/m) dV/dx (1.1)

where m is mass, x is position, t is time, and V is a time independent potential energy.

a) The kinetic and potential energy expressions for a one dimensional harmonic oscillator are

T = (m/2) (dx/dt)² V = kx²/2 (1.2)

where x is the displacement of the oscillator from its equilibrium position and k is the force constant for the oscillator. Use the relationships in 1.2 to substitute into eq 1.1 and obtain the differential equation for the classical harmonic oscillator.

b) Show that

x(t) = A cos(Bt) (1.3)

where A and B are constants, is a solution to the differential equation found in part a of this problem. Find the value for B in terms of k, m, and/or other constants. (The value for A is determined by the initial conditions for the oscillator).

c) For a conservative system we may say

H = T + V = E = constant (1.4)

Use this relationship and the expression for x(t) given in eq 1.3 to find the value for E, the total energy. Give your answer in terms of A, k, m, and/or other constants. Show that the value for E is in fact a constant, that is, independent of time.

2) In a particular study of the photoelectric effect an unknown metal surface was illuminated with light of wavelength l = 254. nm. Electrons were ejected from the metal. The maximum kinetic energy of the ejected electrons was 0.42 eV. Based on this information find F₀ (the work function for the metal, in eV) and l₀ (threshold wavelength for production of electrons, in nm).

3) The Planck distribution law for blackbody radiation is

M(l,T) dl = (2phc²/l⁵) [exp(hc/lkT) - 1]^-1 dl (3.1)

Starting with eq 3.1, do the following:

a) Show that the Planck distribution law reduced to the classical (Rayleigh-Jeans) expression

M(l,T) dl = (2pckT/l⁴) dl (3.2)

in the limit l ® ¥. (Hint: Recall that the Taylor series expansion for e^x is e^x = 1 + x + x²/2! + x³/3! + ...)

b) Find the value of l_max, the wavelength of maximum light intensity, predicted from the Planck distribution law. Based on your result, calculate A_W, the Wein's law constant. (When you get your final expression for A_W you will either have to make a simplifying approximation or find A_W by trial and error.) Recall that the Wien displacement law is:

T l_max= A_W (3.3)

c) Find M(T), the total intensity of light emitted by an ideal blackbody, according to the Planck distribution law. Based on your result find the value for s, the Stefan-Boltzmann constant. Recall than the Stefan-Boltzmann law is:

E = s T⁴ (3.4)

where E is the energy density at temperature T (in Kelvin).

4) To a first approximation the sun can be considered an ideal blackbody emitter of light. The temperature at the surface of the sun is T = 5800. K, and the mean diameter of the sun is d = 1.391 x 10⁶ km.

a) What is the total amount of energy emitted by the sun per unit time (energy/second)?

b) The main source of energy production for the sun is conversion of hydrogen into helium by nuclear fusion. The stoichiometric reaction for the process is (ignoring neutrino production in the reaction, as the mass of a neutrino is small compared to the mass of the other particles involved in the reaction).

4 ¹H⁺ ® ⁴He²⁺ + 2 e⁺ (4.1)

where ¹H⁺ is a hydrogen nucleus (m = 1.00728 amu), ⁴He²⁺ is a helium nucleus (m = 4.00150 amu), and e⁺ is a positron (the antiparticle of an electron, m = 0.00055 amu).

Assuming that all of the energy released by the sun in blackbody radiation comes from reaction 4.1, find the total mass converted into energy per unit time (kg/s) and the total mass of hydrogen converted into helium per unit time (kg/s).

5) Ultraviolet light from the sun in the region 300 - 400 nm passes through the Earth's atmosphere without significant absorption by the ozone layer or other atmospheric gases. This light is responsible for tanning, and is capable of causing skin cancers. Assuming that the sun is an ideal blackbody emitter at a temperature T = 6000. K, the fraction of sunlight in this region is given by the expression

f(300-400nm) = ò₃₀₀⁴⁰⁰ M(l,T) dl/sT⁴ (5.1)

Find f(300-400nm) for the sun. (Hint: The integral in eq 5.1 cannot be done in closed form. The following approximation can be used to evaluate an integral for a function that is slowly varying over the interval in question.)

ò_a^b M(l,T) dl @ M(l,T) Dl (5.2)

where l = (a+b)/2 and Dl = (b-a).)