Thermal Physics - Adiabatic

The adiabatic expansion of an ideal gas is described by the equation PV ^γ = C, where γ and C are constants. The work done by the gas is expanding adiabatically from the state (V_i, P_i) to (V_f, P_f) is equal to

A. P_fV_f
B. ½(P_i + P_f)(V_f  − V_i)
C. (P_fV_f   − P_fV_i)/(1 − γ)
D. P_i(V_f ^1+ γ − V_i ^1+ γ )/(1 + γ)
E. P_f (V_f ^1- γ − V_i ^1- γ)/(1 + γ)

(GR9677 #73)

Solution:

PV ^γ = C
P = CV ^−γ

W = ∫ P dV
= C _Vi∫^V_f  V ^−γ dV
= [C/(−γ + 1)] V ^−γ+1 _Vi|^V_f
= [C/(1 − γ)] (V_f ^1−γ− V_i ^1−γ)

= (V_fCV_f ^−γ− V_i CV_i ^1−γ)/(1 − γ)

= (V_fP_f  − V_i P_i )/(1 − γ)

= (P_fV_f  − P_iV_i )/(1 − γ)

Answer: C

Thermal Physics - Entropy

Questions 71-73

A system in thermal equilibrium at temperature T consists of a large number N₀ of subsystems, each of which can exist only in two states of energy E₁ and E₂, where . In the expressions that follow, k is the Boltzmann constant.

Which of the following is true of entropy of the systems?

A. It increases without limit with T from zero at T = 0
B. It decreases with increasing T
C. It increases from zero at T = 0 to N₀k ln 2 at arbitrarily high temperature
D. It is given by N₀k [⁵/₂ ln T − ln p + constant]
E. It cannot be calculated from the information given

(GR9277 #73)

Solution:

At low T, entropy S(T → 0) = 0
At high T, S = kσ

σ = ln (number of states with the same total energy)

For 2  states of energy, σ = ln 2
S = k ln 2

For N₀ subsystems.
S = N₀k ln 2

Answer: C

Optics - Thin Film

It is necessary to coat a glass lens with a non-reflecting layer. If the wavelength of the light in the coating is λ, the best choice is a layer of material having an index of refraction between those of glass and air and a thickness of

A. λ/4
B. λ/2
C. λ/√2
D. λ
E. 1.5λ

(GR8677 #73)

Solution:

Index of refraction: 

Air to Layer,
phase difference: Δ_a = λ/2

Layer to Glass,
phase difference: Δ_b = 2t + λ/2

The relative shift: Δ = Δ_b − Δ_a =  2t + λ/2 − λ/2 = 2t

Non-reflecting → destructive interference
Δ = (m +½)λ

m = 0 (the thinnest the better)
2t = (0 +½)λ
t = λ/4

Answer: A


Notes: Click HERE for more info on Thin Film.

Classical Mechanics - Newton's Law

For the system consisting of the two blocks shown in the figure, the minimum horizontal force F is applied so that block B does not fall under the influence of gravity. The masses of A and B are 16.0 kilograms and 4.00 kilograms, respectively. The horizontal surface is frictionless and the coefficient of friction between the two blocks is 0.50. The magnitude of F is most nearly

A. 50 N
B. 100 N
C. 200 N
D. 400 N
E. 1,600 N

(GR0177 #73)

Solution:




If B does not fall under the influence of gravity:



And,



Therefore,



Answer: D