Elements of Modern Physics
252
2. For a gas at temperature T, what is the most probable energy of a particle?
What is the approximate fraction of particles which have more then ten
times the average energy of the particles?
3. For producing Balmer absorption lines, a substantial number of the
hydrogen atoms should be in the first excited state. What is the percentage
of the hydrogen atoms which are in the first excited state in the sun’s
photosphere which is at a temperature of about 6000 K, taking into
account the degeneracy of the states?
4. What is the fraction of the hydrogen molecules which are in the first
excited vibrational state at 3000 K? What is the fraction in the second
excited vibrational state?
5. Show that for T >> T D = θ, the total energy of a solid is given by
E =
1
3 1
3
…
8 20
D
D
D
T
T
RT T
T
−
− +
+
and the corresponding specific heat by
C v =
2
1
3 1
...
20
D
T
R
T
−
+
The approximation is quite good even for T ~ T D . Estimate the specific
heat of copper at T = 300 K (T D for copper is 343 K).
6. Obtain an expression for the energy of a 2-dimensional lattice. What is
the specific heat of this lattice for T → ∞ and for T → 0? The situation
is applicable for layer structures such as graphite whose specific heat at
low temperatures is proportional to T
2
.
7. For Cu, the lattice specific heat a low temperature has the behaviour for
C v ~ 4.6 × 10
–5
T
3
J/mol K. Estimate the Debye temperature for Cu.
8. Given that the Debye temperature for diamond is 1860 K, what is the
specific heat of diamond at room temperature?
9. What is the number of excited phonons and the average energy per phonon
at a given temperature? Show that the average goes to
2
3
hv max for T → ∞ and is
proportional to T for T → 0.
10. Show that the average kinetic energy of the electrons emitted in thermionic
emission is 2kT, and that the average value of the square of velocity
perpendicular to the surface is 2kT/m.
11. Calculate the Fermi energy at 0 K, of silver (density 10.5 g/cc, atomic
weight ≈ 107.87) and sodium (density 0.97 g/cc, atomic weight ≈ 22.99),
252
2. For a gas at temperature T, what is the most probable energy of a particle?
What is the approximate fraction of particles which have more then ten
times the average energy of the particles?
3. For producing Balmer absorption lines, a substantial number of the
hydrogen atoms should be in the first excited state. What is the percentage
of the hydrogen atoms which are in the first excited state in the sun’s
photosphere which is at a temperature of about 6000 K, taking into
account the degeneracy of the states?
4. What is the fraction of the hydrogen molecules which are in the first
excited vibrational state at 3000 K? What is the fraction in the second
excited vibrational state?
5. Show that for T >> T D = θ, the total energy of a solid is given by
E =
1
3 1
3
…
8 20
D
D
D
T
T
RT T
T
−
− +
+
and the corresponding specific heat by
C v =
2
1
3 1
...
20
D
T
R
T
−
+
The approximation is quite good even for T ~ T D . Estimate the specific
heat of copper at T = 300 K (T D for copper is 343 K).
6. Obtain an expression for the energy of a 2-dimensional lattice. What is
the specific heat of this lattice for T → ∞ and for T → 0? The situation
is applicable for layer structures such as graphite whose specific heat at
low temperatures is proportional to T
2
.
7. For Cu, the lattice specific heat a low temperature has the behaviour for
C v ~ 4.6 × 10
–5
T
3
J/mol K. Estimate the Debye temperature for Cu.
8. Given that the Debye temperature for diamond is 1860 K, what is the
specific heat of diamond at room temperature?
9. What is the number of excited phonons and the average energy per phonon
at a given temperature? Show that the average goes to
2
3
hv max for T → ∞ and is
proportional to T for T → 0.
10. Show that the average kinetic energy of the electrons emitted in thermionic
emission is 2kT, and that the average value of the square of velocity
perpendicular to the surface is 2kT/m.
11. Calculate the Fermi energy at 0 K, of silver (density 10.5 g/cc, atomic
weight ≈ 107.87) and sodium (density 0.97 g/cc, atomic weight ≈ 22.99),