Physical Chemistry I Homework Set #1

1.1 The Planck radiation law is ρ(λ) = 8πλ 5

hcehc/λkT −1

Using exponential series expansion ex = 1 + x + x2

2!+ ...

to replace the exponent in the Planck radiation law derive the Rayleigh-Jeans equation

ρ(λ) = 8π kTλ 4 , which only applies at long wavelength.

1.2 When irradiated with light of wavelength 2700 Å what is the maximum velocity with which an electron can escape from

a. Bi (work function W = 4.34 eV)?

b. Cr (W = 4.50 eV)?

c. Cu (W = 4.65 eV)?

1.3 In the Davisson and Germer experiment electrons were diffracted from rows of nickel

atoms on the 111 face of a nickel crystal that are spaced by a distance d = 2.15 Å. The

electron energy was 54 eV and the scattering angle θ was 50° and yielded a diffraction

wavelength of 1.65 Å in good agreement with the deBrogile hypothesis. Davisson and

Germer also observed intense scattering from the nickel 111 face with an electron

acceleration voltage of 65 V when the detector was set at a scattering angle of 44°. Does

this result further support the deBrogile hypothesis? Explain.

1.4 The wavefunction ψ = Ax ℓ− x( ) is a good approximation to the true wavefunction for a

particle trapped in a one-dimensional box of length ℓ running from x = 0 to x = ℓ .

Determine the normalization constant, A.

1.5 A particular periodic particle is described in all space as

P(x) = 12b

( -b < x < b )

P(x) = 0 elsewhere

P(x) = P(x + 2a) period of 2a

A Fourier series which adequately represents this particle is

f (x) = ba 2

+ ∑n=1

∞ 2nπ b

sin nπbacos nπ x

a

For a = 4 and b = 1, use EXCEL to plot on the same graph for –a ≤ x ≤ a

(1) P(x)

(2) f(x), which includes the first five non-zero members of the series.

*HINT: sin 4π4

= 0

1.6 The amplitude of waves a(k) contributing to a wavefunction ψ (x) is given at all k by

a(k) = 1b

−b2

≤ k ≤ b2

a(k) = 0 k > b2

Given Euler’s formula for waves eikx = cos kx + isin kx determine the Fourier transform of

a(k) , i.e., find ψ (x) = 1

2πa(k)eikx dk

−∞

∞

∫ . Show that ψ (x) is normalized over the domain

−∞ < x < ∞ .