Classical Mechanics - Fluid

An open-ended U-tube of uniform cross-sectional area contains water (density 1.0 gram/cm³) standing initially 20 cm from the bottom in each arm. An immiscible liquid of density 4.0 gr/cm³ is added to one arm until a layer 5 cm high forms. What is the ratio h₂/h₁ of the heights of the liquid in the two arms?

A. 3/1
B. 5/2
C. 2/1
D. 3/2
E. 1/1

(GR9677 #30)

Solution:

Given:
ρ_water = 1 gr/cm³
ρ_liquid = 4 gr/cm³
h_a = 5 cm

P₁ = P₂
ρ_lgh_a = ρ_wgh_b 
h_b = ρ_lh_a / ρ_w = (4)(5) / (1) = 20

h₁ = h + h_a  
h₂ = h + h_b 
-------------- −
h₁ − h₂ = h_a − h_b  = 5 − 20 = − 15 (Eq.1)

Initially, h₁ + h₂ = 20 + 20 = 40
Finally, h₁ + h₂ = 40 + 5 = 45 (Eq.2)

(Eq.1) → h₁ − h₂ = − 15
(Eq.2) → h₁ + h₂ = 45
------------------------------ +
2h₁ = − 15 + 45 = 30
h₁ = 15

h₂ = 45 − 15 = 30

h₂/h₁ = 30/15 = 2

Answer: C

Nuclear & Particle Physics - Positronium

Given that the binding energy of the hydrogen atom ground state is E₀ = 13.6 eV, the binding energy of n = 2 state of positronium (positron-electron system) is

A. 8E₀
B. 4E₀
C. E₀
D. E₀/4
E. E₀/8

(GR9277 #30)

Solution:

Energy levels of Positronium is half those of Hydrogen (See GR8677 #99)

E_n(H)  = − 13.6 / n²
E_n(Ps) = ½ E_n(H)  

For n = 2,

E_2(Ps) = ½ × ( − 13.6 / 2²)  = − E₀/8

Answer: E

Nuclear & Particle Physics - Electron Configuration

The configuration of the potassium atom in its ground state is 1s² 2s² 2p⁶ 3s² 3p⁶ 4s¹. Which of the following statement about potassium is true:

A. Its n = 3 shell is completely filled
B. Its 4s subshell is completely filled
C. Its least tightly bound electron has l = 4
D. Its atomic number is 17
E. Its electron charge distribution is spherically symmetrical

(GR8677 #30)

Solution:

A. FALSE.
n = 3 → l = 0, 1, 2 → s, p, d subshells, but the configuration does not have d subshells.

B. FALSE
4s subshell → for s subshell total electron should be 2 not 1.

C. FALSE
l = 0, 1, 2, 3, 4… correspond to s, p, d, f, g, … subshells, but the configuration does not have g subshells.

D. FALSE

the number of electrons in the configuration = 2+2+6+2+6+1 = 19, not 17.

E. TRUE
The single s-orbitals (l = 0) are shaped like spheres. For n = 1 the sphere is "solid" (it is most dense at the center and fades exponentially outwardly), but for n = 2 or more, each single s-orbital is composed of spherically symmetric surfaces which are nested shells (i.e., the "wave-structure" is radial, following a sinusoidal radial component as well).

The outermost electron shell (valence shell) is s¹ with n >1 and l = 0 → spherically symmetric.

Also note that Potassium has one outer electron, like Hydrogen. So it will also have a spherically symmetrical charge distribution.

Answer: E

Quantum Mechanics - Wave Function

Which of the following functions could represent the radial wave function for an electron in an atom? (r is the distance of the electron from the nucleus; A and b are constants.)
I. Ae^−br
II. A sin (br)
III. A/r

A. I only
B. II only
C. I and II only
D. I and III only
E. I, II, and III

(GR0177 #30)

Solution:

Property of a wave function: Ψ is finite.
Ψ(r → ∞) and Ψ(r → 0) cannot be ∞.

Ψ(r → ∞) = Ae^−∞ → 0 
Ψ(r → 0) = Ae⁻⁰ → A
I. TRUE

Ψ(r → ∞) = A sin ∞ ↛ 0.
II. FALSE

Ψ(r → 0) = A/0 → ∞.
III. FALSE

Answer: A