Question

Consider 1 mole of each of two Einstein solids, A and B. The oscillator frequency in B is 50% greater than that in A, and Cv for A at 100K is 7.60 J/K mol. At what temperature will Cv for B have this value?

Answer #1

For the Einstein harmonic oscillator model of the solid, if the
oscillators have angular frequency ω = 107 s
−1, how much
heat must be supplied to a mole of crystal in order to raise its
temperature by 10−2 degrees in the vicinity of 1K, and
in the vicinity of 100K?

A mole of two-state systems each has energy difference 2.346 ×
10-21 J.
1)
At what temperature are 90% of the two-state parts in their
low-energy states?
T90%=
K
2)
What is the heat capacity at T= 17 K?
CV=
J/K
3)
What is the heat capacity at T= 85 K?
CV=
J/K
4)
What is the heat capacity at T= 170 K?
CV=
J/K

A mole of two-state systems each has energy difference
2.346 × 10-21 J.
1)
At what temperature are 90% of the two-state parts in
their low-energy states?
T90%=
K
2)
What is the heat capacity at T= 17 K?
CV=
J/K
3)
What is the heat capacity at T= 85 K?
CV=
J/K
4)
What is the heat capacity at T= 170 K?
CV=
J/K

Consider a system of two Einstein solids, A=20 and B=20, sharing
a total of 100 units of energy. Assume that the solids are weakly
coupled, and that the total energy is fixed.
How many different macrostates are available to this system?
How many different microstates are available to this system?
Assuming that this system is in thermal equilibrium, what is the
probability of finding all the energy in solid A?
What is the probability of finding exactly half of the...

A mole of two-state systems each has energy difference
2.346 × 10-21 J.
1)
What is the heat capacity at T= 17 K?
CV=
J/K
2)
What is the heat capacity at T= 85 K?
CV=
J/K
3)
What is the heat capacity at T= 170 K?
CV=
J/K

16. Consider two boxes that are separated. Both boxes are 1-m3.
The first box is filled with 1 mole of N2 at 100K and the second
box is filled with 1 mole of Ar at 200K. The two boxes are placed
into contact so heat can transfer between them. What is the final
temperature?
a. 137.5 K b. 150.0 K c. 162.5 K
17. Now suppose the wall between the two boxes is removed after
they have equilibrated. What is...

A small Einstein solid consists of 2000 atoms, each free to
oscillate in three dimensions. The solid contains a total of 30 000
units of energy.
(a) Calculate the entropy of the solid.
(b) If the temperature of the solid is 650 K, what is the
approximate value of one unit of energy (in J)?
(c) What is the resonant frequency of the oscillators in the
Einstein solid?

A system of 6 identical 1-dimensional harmonic oscillators (it
could be two atoms of an Einstein model solid). What is the
probability that the first oscillator has 2 quanta of vibrational
energy when the system has 3 quanta of vibrational energy?
Hint: You may find it useful to consider how many microstates
the system has and how many microstates the rest of the system has
when the first oscillator has 2 quanta.
Answer is 5/56
For the system considered in...

. Consider two systems,1and 2,interacting thermally but not
mechanically or diffusively. Their initial temperatures are 0 ◦C
and 50 ◦C, respectively. The heatcapacitiesofsystem1areCV1 =2R,Cp1
=2.8R,andthoseofsystem2 areCV2 =3R,Cp2 =4R; R, the gas
constant=8.31 J/(K mole). Find (a) their temperature after reaching
equilibrium, (b) the change in entropy of system 1, (c) The change
in entropy of the total combined system. (d) Why is one of the heat
capacities (CV orCp) irrelevant here?

(a) In an experiment to produce a Bose-Einstein condensate,
approximately 107 atoms of a bose-Einstein condensate were found to
be present.
The rubidium 87Rb particles have cooled down to a temperature of
200 nK. The atoms were confined in a
volume of about 10-15 m3. Was there condensation? 87Rb: M = 86,909
g/mol, 1,44316 × 10-25 kg,
spin 1 or 2 (depending on the orientation of the nucleus spin
relative to the rest of the atom).
(b) The molecular molecular...

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