Prove that for a polymer solution in the athermal limit (ΔHm = 0), no phase separation...
Prove that for a polymer solution in the athermal limit (ΔHm = 0), no phase separation can occur in the Flory-Huggins model. (Hint: ΔGm < 0 means that mixing is spontaneous, but it does no say by itself whether there is one phase or two.)
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Prove that for a polymer solution in the athermal limit (ΔHm =
0), no phase separation...
11. One-dimensional polymer, consider a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain....
11. One-dimensional polymer, consider a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain. Each molecule can align along either its long axis (of length 2a) or short axis (length a). The energy of the monomer aligned along its shorter axis is higher by e, that is, the total energy is H = 0, where U is the number of monomers standing up. (a) Calculate the partition function, Z(T, N), of the polymer. (b) Find the relative probabilities...
Due date: April 7, 2020 200 Solubaty limit- Homework #7 (Chapter 9: Phase Diagrams) 150 Liquid...
Due date: April 7, 2020 200 Solubaty limit- Homework #7 (Chapter 9: Phase Diagrams) 150 Liquid solution Ganes Solution sugar 100 1 50 1. Consider the Cu-Ni binary Liquid solution (syrup) phase diagram (see the following old figure) and answer the questions: A copper-nickel alloy of composition 70 wt% Ni-30 wt% 207 Cu is slowly heated from a temperature of 1300°C. Il (a) At what temperature does the Sugar 60 80 Water 100 first liquid phase form? (b) What is...
Exercise 4.1.6: Sketch a phase portrait that shows an unstable limit cycle. The image I included ...
Exercise 4.1.6: Sketch a phase portrait that shows an unstable
limit cycle.
The image I included below is meant to help with the exercise
above. It is the background information given.
This new kind of friction then gives us a new differential equation A simulation of this equation results in the trajectories shown in Figure 4.8, right. spring friction Figure 4.8: Upper: spring force and friction force for the Rayleigh clarinet model. Lower Left: Two representative trajectories for this model....
Let p be a prime number. Prove that if there exists a solution to the congruence...
Let p be a prime number. Prove that if there exists a solution to the congruence c(mod p), then there exist integers m, n such that p = m2 + 2n2. Hint. Make a careful study of the proof of Fermat's Two Square Theorem, and then try to modify that proof (or at least some portion of it) to come up with a proof of this statement. Toward the end of the proof, the following observation can be helpful: If...
2. Find the solution set of the linear program > 0 (a) by the two-phase method....
2. Find the solution set of the linear program > 0 (a) by the two-phase method. (b) by solving the dual program with the simplex method and then using the "comple- mentary slackness" conditions to solve the original problem (c) by the dual simplex method. Show how the same results can be obtained by each of the three methods. (You will find that the problem has more than one minimizer.)
We have the following Limit Comparison Test for improper integrals: Theorem. Suppose f(x), g(x) are two...
We have the following Limit Comparison Test for improper integrals: Theorem. Suppose f(x), g(x) are two positive, decreasing functions on all x > 1, and that lim f(x) =c70 x+oo g(x) Then, roo 5° f(x) dx < oo if and only if ſº g(x) dx < 00 J1 (a) Using appropriate convergence tests for series, prove the Limit Comparison Test for improper integrals. (Hint: Define two sequences an = f(n), bn = g(n). What can you say about the limit...
LAB 2A SEPARATION OF LIQUIDS BY SIMPLE DISTILLATION ANALYSIS BY GAS CHROMATOGRAPHY LAB 2A Pre-Lab Questions...
LAB 2A SEPARATION OF LIQUIDS BY SIMPLE DISTILLATION ANALYSIS BY GAS CHROMATOGRAPHY LAB 2A Pre-Lab Questions NAME DATE LAB SECTION Answer the following questions in the pre-lab section of your notebook. 1. Which compound, acetone or 1-propanol, has the higher boiling point? Why? 2. Do you expect acetone to be soluble in 1-propanol? Why or why not? Model: For this question, you will be graded on completion, rather than correctness of the model. In the post-lab, you will be asked...
Two in-phase loudspeakers are placed along a wall and are separated by a distance of 6.00 m. They...
Two in-phase loudspeakers are placed along a wall and are
separated by a distance of 6.00 m. They emit sound (take vs = 343
m/s) with a frequency of 137.2 Hz. A person is standing away from
the wall, in front of one of the loudspeakers. What is the closest
distance x from the speaker the person can stand and hear a sound
intensity maximum?
4. [5] Two in-phase loudspeakers are placed along a wall and are separated by a...
Nicotine |H2O T u210 P 2 61 Tm P 1 0 0.2 0.4 0.6 0.8 1 Mole fraction of nicotine, X Temperature, 0/°C The phase diag...
Nicotine |H2O T u210 P 2 61 Tm P 1 0 0.2 0.4 0.6 0.8 1 Mole fraction of nicotine, X Temperature, 0/°C The phase diagram for nicotine and water is shown here. Note the two boundaries for pure water (left) and pure nicotine (right) and mole fractions of nicotine across the horizontal axis. The letter "P" means phases, and this diagram has two regions: single-phase (one liquid, fully dissolved solid), and the other two-phase (two liquids) Here, the curve...
11. One-dimensional polymer, consider a polymer formed by connecting N disc-shaped molecules into a one-dimensional chain. Each molecule can align along either its long axis (of length 2a) or short axis (length a). The energy of the monomer aligned along its shorter axis is higher by e, that is, the total energy is H = 0, where U is the number of monomers standing up. (a) Calculate the partition function, Z(T, N), of the polymer. (b) Find the relative probabilities...
Due date: April 7, 2020 200 Solubaty limit- Homework #7 (Chapter 9: Phase Diagrams) 150 Liquid solution Ganes Solution sugar 100 1 50 1. Consider the Cu-Ni binary Liquid solution (syrup) phase diagram (see the following old figure) and answer the questions: A copper-nickel alloy of composition 70 wt% Ni-30 wt% 207 Cu is slowly heated from a temperature of 1300°C. Il (a) At what temperature does the Sugar 60 80 Water 100 first liquid phase form? (b) What is...
Exercise 4.1.6: Sketch a phase portrait that shows an unstable
limit cycle.
The image I included below is meant to help with the exercise
above. It is the background information given.
This new kind of friction then gives us a new differential equation A simulation of this equation results in the trajectories shown in Figure 4.8, right. spring friction Figure 4.8: Upper: spring force and friction force for the Rayleigh clarinet model. Lower Left: Two representative trajectories for this model....
Let p be a prime number. Prove that if there exists a solution to the congruence c(mod p), then there exist integers m, n such that p = m2 + 2n2. Hint. Make a careful study of the proof of Fermat's Two Square Theorem, and then try to modify that proof (or at least some portion of it) to come up with a proof of this statement. Toward the end of the proof, the following observation can be helpful: If...
2. Find the solution set of the linear program > 0 (a) by the two-phase method. (b) by solving the dual program with the simplex method and then using the "comple- mentary slackness" conditions to solve the original problem (c) by the dual simplex method. Show how the same results can be obtained by each of the three methods. (You will find that the problem has more than one minimizer.)
We have the following Limit Comparison Test for improper integrals: Theorem. Suppose f(x), g(x) are two positive, decreasing functions on all x > 1, and that lim f(x) =c70 x+oo g(x) Then, roo 5° f(x) dx < oo if and only if ſº g(x) dx < 00 J1 (a) Using appropriate convergence tests for series, prove the Limit Comparison Test for improper integrals. (Hint: Define two sequences an = f(n), bn = g(n). What can you say about the limit...
LAB 2A SEPARATION OF LIQUIDS BY SIMPLE DISTILLATION ANALYSIS BY GAS CHROMATOGRAPHY LAB 2A Pre-Lab Questions NAME DATE LAB SECTION Answer the following questions in the pre-lab section of your notebook. 1. Which compound, acetone or 1-propanol, has the higher boiling point? Why? 2. Do you expect acetone to be soluble in 1-propanol? Why or why not? Model: For this question, you will be graded on completion, rather than correctness of the model. In the post-lab, you will be asked...
Two in-phase loudspeakers are placed along a wall and are
separated by a distance of 6.00 m. They emit sound (take vs = 343
m/s) with a frequency of 137.2 Hz. A person is standing away from
the wall, in front of one of the loudspeakers. What is the closest
distance x from the speaker the person can stand and hear a sound
intensity maximum?
4. [5] Two in-phase loudspeakers are placed along a wall and are separated by a...
Nicotine |H2O T u210 P 2 61 Tm P 1 0 0.2 0.4 0.6 0.8 1 Mole fraction of nicotine, X Temperature, 0/°C The phase diagram for nicotine and water is shown here. Note the two boundaries for pure water (left) and pure nicotine (right) and mole fractions of nicotine across the horizontal axis. The letter "P" means phases, and this diagram has two regions: single-phase (one liquid, fully dissolved solid), and the other two-phase (two liquids) Here, the curve...
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