 |
|
|
|
|
|
|
Helium (He) at a rate of 1.5 lbm/s is compressed in an adiabatic compressor from an initial condition of 70 F, 14.7 Psia to a final condition of 1070 F, 147 Psia.
Determine:
a) The power required to run this compressor,
b) The isentropic efficiency of this compressor,
c) The T-s diagram (not-to-scale), and
d) The entropy generation for this process.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 155306
|
$ 2.0 |
|
|
|
|
The following example is extracted from Knight’s Physics for Scientists and Engineers
1. A heat engine follows the ideal-gas process shown in Fig.21-101.
Assume the gas is monatomic. Analyze this engine to determine:
(a) the net work done per cycle;
(b) the engine’s thermal efficiency;
(c) the engine’s power output if it runs at 600 rpm .
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 155518
|
$ 2.0 |
|
|
|
|
The following example is extracted from Knight’s Physics for Scientists and Engineers
A heat engine with a diatomic gas as the working substance uses the closed cycle shown in Fig.21-102.
How much work does this engine do per cycle and what is its thermal efficiency? ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 155520
|
$ 1.5 |
|
|
|
|
The following example is extracted from Knight’s Physics for Scientists and Engineers
A refrigerator using helium gas operates on a reversed Brayton cycle with a pressure ratio of 5.0.
Prior to compression, the gas occupies 100cm3 at a pressure of 150kPa and a temperature of -23°C .
Its volume at the end of the expansion is 80cm3 .
What are the refrigerator’s coefficient of performance and its power input if it operates at 60 cycles per second?
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 155522
|
$ 1.5 |
|
|
|
|
Heat Pump
In the winter you probably want to heat your home. Although we traditionally think of heat as only flowing from hot to cold (e.g., from inside your cozy home to the wintry outside), we also know that we can get heat to flow the other way, by doing work. The device that does this is known as a “heat pump”.
Consider that your house is at 22°C, and the temperature outside is 0°C. Previously in the Window problem, we showed that if you have a poorly insulated window, you could have a heat leak (under these temperature conditions) of 1500 watts. Here we will explore how much work you would need to do to keep your home at 22°C.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 156274
|
$ 2.0 |
|
|
|
|
Jumping Ball
Imagine that we have a solid spherical ball of aluminum sitting on the table. If all the thermal energy were instantly converted into center-of-mass kinetic energy, and Vcm were directed straight up, how high would the ball go?
Consider step by step way for the answer:
a) Using the equipartition theorem for atoms in the solid acting as three-dimensional harmonic oscillators, give a formula for the average energy of each atom. (Hint: Remember the relationship between temperature and each quadratic term in the energy of a particle.)
b) What is the thermal energy of a 1-kg ball at 300 K?
c) Compute the height to which the ball would rise, if all this thermal energy were instantly converted into upward center-of-mass motion.
d) Why don’t objects normally rise by themselves in this way?
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 156275
|
$ 2.0 |
|
|
|
|
Problem #1: Black Hole Entropy
[15 pts]
A black hole has the strange property that it is completely determined by its mass, ie, two black holes of the same mass axe physically indistinguishable.
Let's say you throw something into the black hole. In order for the second law of thermodynamics to hold, it must be the case that the black hole's entropy increases by at least the amount of entropy in the object you threw in. But because of this strange property of black holes, this must be independent of what you throw in.
In other words, a black hole of mass M must have a higher entropy than any other system of mass M. So a good way to estimate the entropy of a black hole of mass M is to calculate the most entropic configuration of ordinary matter of mass M. One particularly entropic form of matter is light.
[1 pts] (a) What is the energy (not entropy) of N photons of wavelength ? Find the equivalent mass, using E = Mc2.
Now, it turns out that for a wide variety of systems, the entropy of the system is roughly proportional to the number of particles. So the most entropic system with a given mass M would consist of particles of the smallest possible energy (including rest mass). The best candidate would be photons of the greatest possible wavelength (and so lowest energy).
[2 pts] (b) Assuming the wavelength can be no greater than the radius R of the black hole, use part (a) and the approximation S ~ kBN to find the greatest possible entropy of a system of photons of mass M. This is a good estimate for the entropy of a black hole of mass M.
[2 pts] (c) A black hole of mass M can be thought of as a sphere of radius R = 2GM/c^2 where G is the gravitational constant and с is the speed of fight. Show that the entropy you found in (b) is proportional to the area of the surface of the black hole.
[3 pts] (d) Using E = Mc2, we can rewrite the entropy as a function of energy. Using this, compute the temperature of the black hole.
[2 pts] (e) A black hole is a perfect black body radiator. Using the temperature from (d), find the total power emitted by the black hole.
[5 pts] (f) The radiation leaving a black hole carries energy (mass), and so without further input, a black hole will eventually radiate away all its mass. Determine the time this will take for a black hole of mass M. What is this time for a solar mass black hole? What about for M = 14 TeV/c2 = 2.5 x10-23 kg, the mass of potential LHC black holes?
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158540
|
$ 6 |
|
|
|
|
The conduction electrons in insulators and semiconductors can be described by a Fermi gas with a density of states equal to zero in
certain regions called energy gaps. Electrons cannot have energies in these regions. In this problem, we consider a simplified model of a system of degenerate, non-interacting electrons, in which the density of states is shown in the figure below.
[3 pts] (a) Show that the probability of finding an electron in a state with energy above the chemical potential is the same as the probability of finding an electron absent from a state with energy below (i.e. the probability of having no electron with energy below ), at any given temperature .
[8 pts] (b) Suppose that the density of states D(e) is given by (see figures):
Suppose that at = 0, all states with e < —eo are occupied while the other states are empty. For > 0, some states with e > eo will be occupied while some states with e < - eo will be empty. Assuming the number of particles is the same in both cases, what is the chemical potential ()?
[HINT: Do NOT integrate. Find a change of variables that makes the two integrals look similar.]
[4 pts] (c) If there is an excess of Nd electrons that are accommodated by the states with e > 0, find ( = 0) in terms of eo, Nd, and a?
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158543
|
$ 6 |
|
|
|
|
Many connections have been made between entropy and information theory. Shannon (1949) did some pioneering work in this regard by realizing that systems with high entropy can be related to lack of information. He made the connection by realizing that for a system in contact with a heat bath at temperature r (Canonical Ensemble) the entropy can be written fig.21-108
where Pr is the probability that the system is in state r. Thus when many, many of the Pr's are small but non-zero (e.g. at high temperature) we have little information about the state of the system and the entropy is large. Derive the above expression assuming a system with non-degenerate accessible states E1, E2, …, E in contact with a heat bath at temperature r.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158545
|
$ 3.0 |
|
|
|
|
The energy levels of a quantum mechanical, three-dimensional rigid rotor of moment of inertia I are given by
EJ,m = h2J(J + 1)/2I,
where J = 0,1,2,... is the angular momentum quantum number, and M = — J, —J + 1, J is the spin quantum number. Consider a system of N distinguishable quantum mechanical rotors in contact with a heat bath at temperature r:
i) Find an expression for the thermodynamical internal energy of the system.
[Hint: There is degeneracy in this problem]
ii) Under what conditions can the sum in part (i) be approximated by an integral?
In this case calculate the specific heat Cv of the system.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158546
|
$ 3.0 |
|
|
|
|
Chemical potential and Gibbs sum. Consider a system free to exchange energy and particles with a bath at temperature and chemical potential .
[4 pts] (a) When the system is free to exchange only energy with the bath, we learned that the equilibrium energy of the system corresponds to the minimum of the free energy F(, N,V;U)= U — with respect to energy variations. When the system can exchange particles as well as heat with the bath, we define a different free energy Ф (, , V;U,N) = U — — N. Show that Ф is minimized with respect to energy and particle variations at equilibrium.
[3 pts] (b) Show that (up to a constant multiplicative factor which doesn't matter), the Gibbs sum is given by Eqv21-110:
Why is the constant factor insignificant?
[4 pts] (c) Argue that the sum is dominated by a single term (corresponding to the equilibrium configuration) and therefore find a formula for Ф(, , V) - the equilibrium value of Ф(, , V; U, N) - in terms of the Gibbs sum.
[4 pts] (d) Consider a third potential O(,,P) = Ф(, , V) + PV. Evaluate O for the ideal gas.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158548
|
$ 6 |
|
|
|
|
1. Answer the following questions qualitatively:
1.1. (5 pts) The Helmholtz free energy, F, is minimum for a system (with fixed V and N) in thermal equilibrium with a bath at temperature T. How is this statement consistent with the second law?
1.2. (5 pts) Briefly explain what is meant by micro-canonical, canonical and grand canonical ensembles.
1.3. (5 pts) Explain the Gibbs Paradox.
1.4. (5 pts) Briefly explain the symmetrization postulate and its importance to statistical mechanics.
1.5. (5 pts) Describe the Einstein and Debye models of heat capacity and what each model is able to explain about experimental phenomena. ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158549
|
$ 6.0 |
|
|
|
|
10 points). At 1 atmosphere (760 Torr) Mercury boils at 357 degrees C. It has a vapor pressure of 0.0017 Torr at 25 °C. Assuming that Mercury vapor behaves as an ideal gas, what is the latent heat of vaporization (per mole) for Mercury. ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158550
|
$ 3.0 |
|
|
|
|
3. (15 Points) A cup of hot water (surface area "A'') sits in a large room and cools through evaporation.
Assume that the water vapor behaves as an ideal (monatomic) gas, and compute the rate of energy loss (dE/dt) through evaporation at a given temperature.
For the calculation assume that an equilibrium vapor exists at the water interface.
As an approximation, assume that half of the molecules in this interfacial layer move at their thermal velocity (given by the equi-partition theorem) directly away (normal) from the water-air interface (this will tend to slightly over estimate the energy loss rate).
Also, assume that these lost molecules are replenished by evaporation.
In addition to the water temperature, T, express your result using the following parameters:
h (latent heat per water molecule),
m (mass of water molecules),
TB (boiling temperature of water at atmospheric pressure,
РA), and kB (Boltzmann's constant).
Be sure to explain the steps in the analysis. ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158551
|
$ 4.0 |
|
|
|
|
4. (10 points) The molecules of a certain gas have two internal energy states (E1=0 and E2 = e0). The molecules are non-interacting and otherwise ideal.
Compute the thermal energy and heat capacity of a gas of N such molecules at temperature T.
Define a characteristic temperature, TC , in terms of e0 and write the Energy and Heat capacity in the corresponding low (T« TC) and high temperature (T»TC) limits. ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158552
|
$ 4.0 |
|
|
|
|
(15 Points) By considering the dependence of g (specific Gibbs free energy) on temperature, show that in going from a low temperature phase to a high temperature phase of a substance that the latent heat must always be positive (i.e., heat must be absorbed to enable the phase transition). ...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158553
|
$ 4.0 |
|
|
|
|
(a) (15 points) An insulated box has N compartments that are themselves insulated from each other and which contain different (monatomic) ideal gases at the same pressure. All other quantities (Nk, Tk, Vk for the кth compartment) are different. If all of the partitions between compartments are suddenly removed, what is the new equilibrium pressure, temperature, and the change in entropy.
Express the final temperature in terms of the initial temperatures,
(b) (10 points) Now suppose that the temperature is initially the same in the isolated compartments, while all other parameters are different (i.e., Nk, Pk, Vk for the kth compartment). What will be the equilibrium temperature, pressure, and entropy change once the partitions are removed.
Express the final pressure in terms of the initial pressures.
...MORE
Subject: Physics | Topic: Thermodynamics | ID: 158554
|
$ 6.0 |
|
|
