Thermal Physics - Power

A 100-watt electric heater element is placed in a pan containing one liter of water. Although the heating element is on for a long time, the water, though close to boiling does not boil. When the heating element is removed, approximately how long will it take the water to cool by 1 degree Celsius? (Assume that the specific heat for water is 4.2 kJ/kg^oC)

A. 20 s
B. 40 s
C. 60 s
D. 130 s
E. 200 s

(GR9677 #13)

Solution:

P = 100 W
V = 1 L = 1 m³ → m = 1 kg (STP)
ΔT = 1^oC
c = 4.2 kJ/kg^oC

Q = mcΔT = Pt
1 × 4200 × 1 = 100t
t = 42 s

Answer: B

Thermal Physics - Specific Heat

Two identical 1 kg blocks of copper metal one initially at a temperature T₁ = 0^oC and the other initially at a temperature T₂ = 100^oC are enclosed in a perfectly insulating container. The two blocks are initially separated. When the blocks are placed in contact, they come to equilibrium at a final temperature T_f. The amount of heat exchanged between the two blocks in this process is equal to which of the following? (The specific heat of copper metal is equal to 0.1 kilocalorie/kg^oK)

A. 50 Kcal
B. 25 Kcal
C. 10 Kcal
D. 5 Kcal
E. 1 Kcal

(GR9677 #14)

Solution:

Q = mcΔT
|Q|_gain = |Q|_lost 
m₁c₁ΔT₁  = m₂c₂ΔT₂

m₁ = m₂ = 1 kg
c₁ = c₂ = c_copper = 0.1 kcal/kg^oK

T₁ = 0^oC = 273^oK

T₂ = 100^oC = 373^oK

ΔT₁ = ΔT₂
T_f   − T₁ = T₂ − T_f 
T_f  = (T₂ + T₁)/2 = (100 + 0)/2 = 50^oC = 323^oK

|Q|_gain = |Q|_lost  = m₁c₁ΔT₁  = 1 × 0.1 × (323 − 273) = 5kcal

Answer: D

Thermal Physics - Specific Heat

For an ideal gas, the specific heat at constant pressure C_p is greater than the specific heat at constant volume C_v because the

1. Gas does work on its environment when its pressure remains constant while its temperature is increased.
2. Heat input per degree increase in temperature is the same in processes for which either the pressure or the volume is kept constant.
3. Pressure of the gas remains constant when its temperature remains constant.
4. Increase in the gas’s internal energy is greater when the pressure remains constant than when the volume remains constant
5. Heat needed is greater when the volume remains constant than when the pressure remains constant.

(GR8677 #14)

Solution:

(A) TRUE.
Heat Capacity: C = Q/dT

First law of Thermodynamics: the change in internal energy of a system dU is equal to the heat Q added and the work, W done on or by the system 
→ dU = Q ± W

W done on the system → +W
W done by the system → −W

Gas (the system) does work on its environment
W done by the system
dU = Q − W

At constant V:
Work, W = PdV = 0
Q = dU
C_v = dU/dT

At constant P:
Work, W = PdV ≠ 0
Q = dU + W
C_p = dU/dT + PdV/dT = C_v + PdV/dT
C_p  C_v

(B) FALSE.
This means C_p = C_v, but according to A, C_p  C_v

(C) FALSE.
Ideal gas law: PV = NkT
If T constant, P changes if V changes.

(D) FALSE.
Heat Capacity, C = Q/dT does not depend on the gas’ internal energy, U

(E) FALSE.
See A. At constant V, Q = dU.
At constant P, Q = dU + W.

Answer: A

Nuclear & Particle Physics - Spin

The hypothesis that an electron possesses spin is qualitatively significant for the explanation of all the following topics EXCEPT the

A. Structure of the periodic table
B. Specific heat of metals
C. Anomalous Zeeman effect
D. Deflection of moving electron by a uniform magnetic field
E. Fine structure of atomic spectra

(GR8677 #27)

Solution:

(A) TRUE
Spin → Pauli exclusion principle → electron configuration → Structure of the periodic table

(B) TRUE
Specific heat for Fermions (½-integer spin) is different from Bosons (integer spin).

(C) TRUE
Zeeman Effect: the splitting of spectral lines when an external magnetic field is applied.
"Normal" Zeeman effect → This type of splitting is observed for spin 0 states since the spin does not contribute to the angular momentum.
"Anomalous" Zeeman effect → When electron spin is included, there is a greater variety of splitting patterns.

(D) FALSE
Deflection of moving electron by a uniform magnetic field does not depend on spin

(E) TRUE
Fine structure = the splitting of the spectral lines of atoms due to quantum mechanical (electron spin) and relativistic corrections.

Answer: D

Thermal Physics - Einstein solid

One feature common to both the Debye theory and the Einstein theory of the specific heat of a crystal composed of N identical atoms is that the

1. Average energy of each atom is 3kT
2. Vibrational energy of the crystal is equivalent to the energy of 3N independent harmonic oscillators
3. Crystal is assumed to be continuous for all elastic waves
4. Speed of the longitudinal elastic waves is less than the speed of the transverse elastic waves
5. Upper cutoff frequency of the elastic waves is the same

(GR8677 #51)

Solution:

Einstein solid:

• Thermal properties of crystal
• Highly idealized model focusing only the vibrational modes of crystal
• Assumption: Each of N atoms is free to vibrate around its equilibrium position in any of 3 coordinate directions by a harmonic force with a natural frequency ω₀.

Einstein solid equipartition energy:

• Number of oscillator N = N₀,
• 3-D Harmonic Oscillator, f = 6
• U = (6/2) N₀kT = 3N₀kT

Answer: B

Thermal Physics - Diatomic Molecule

In a gas of N diatomic molecules, two possible models for a classical description of a diatomic molecule are:



Which of the following statements about this gas is true?

A. Model I has a specific heat c_v = ³⁄₂Nk
B. Model II has a smaller specific heat than Model I
C. Model I is always correct
D. Model II is always correct
E. The choice between Models I and II depends on the temperature

(GR8677 #87)

Solution:

Equipartition of Energy: E = ½ fNkT
Specific heat: c_v = ∂E/∂t = ½ fNk

f = Degree of Freedom (DoF)

(A) FALSE
Model I is diatomic molecules in low temperature
→ 5 DoF (3 translational + 2 rotational)
→ c_v = ⁵⁄₂Nk

(B) FALSE
Model II is diatomic molecules in high temperature
→ 7 DoF (3 translational + 2 rotational + 2 vibrational)
→ c_v = ⁷⁄₂Nk

(C) and (D) are vague answers.

Answer: E

Thermal Physics - Superconductor

Which of the following curves is characteristic of the specific heat C_v of a metal such as lead, tin, or aluminum in the temperature region where is becomes superconducting?

(GR9677 #95)

Solution:




Answer: E