This is supplemental Material from "Molecular Driving Forces, K.A. Dill, S. Bromberg", Garland Science, New York:NY, 2003, Chapter 1. See the next three screencasts. This content is useful for graduate level courses that go into more depth or for students interested in more background on probability.
Principles of Probability I, General Concepts, Correlated and Conditional Events. (msu.edu, 17min) (Flash)
Comprehension Questions:
1. Estimate the probability of pulling an king from a randomly shuffled deck of 52 cards.
2. A coin is flipped 5 times. Estimate the probability that heads is observed three of the 5 times.
3. A die (singular of dice) is a cube with the numbers 1-6 inscribed on its 6 faces. If you roll the die 7 times, what is the probability that 5 will be observed on all 7 rolls?
Principles of Probability II, Counting Events, Permutations and Combinations. This part discusses the binomial and multinomial coefficients for putting particles in boxes. The binomial and multinomial coefficient are used in section 4.2 to quantify configurational entropy. (msu.edu, 16min) (Flash) You might like to check out the sample calculations below before attempting the comprehension questions.
Comprehension Questions:
1. Write the formulas for the binomial coefficient, the multinomial coefficient, and the multinomial with repetition.
2. Ten particles are distributed between two boxes. Compute the number of possible ways of achieving 7 particles in Box A and 3 particles in Box B.
3. Note that the binomial distribution is a special case of the multinomial distribution where the number of categories is 2. Also note that the total number of events for a multinomial distribution is given by M^N where M is the number of categories (aka. outcomes, e.g. boxes) and N is the number of objects (aka. trials, e.g. particles). The probability of a particular observation is given by the number of combinations divided by the total number of events. Compute the probability of observing 7 particles in Box A and 3 Particles in Box B.
4. Ten particles are distributed between three boxes. Compute the probability of observing 7 particles in Box A, 3 particles in Box B, and zero particles in Box C.
5. Ten particles are distributed between three boxes. Compute the probability of observing 3 particles in Box A, 3 particles in Box B, and 4 particles in Box C.
Principles of Probability III, Distributions, Normalizing, Distribution Functions, Moments, Variance. This screencast extends beyond material covered in the textbook, but may be helpful if you study statistical mechanics in another course. (msu.edu, 15min) (Flash)
Connecting Microstates, Macrostates, and the Relation of Entropy to Disorder (uakron.edu, 14min). For small systems, we can count the number of ways of arranging molecules in boxes to understand how the entropy changes with increasing number of molecules. By studying the patterns, we can infer a general mathematical formula that avoids having to enumerate all the possible arrangements of 10^23 molecules (which would be impossible within several lifetimes). A surprising conclusion of this analysis is that entropy is maximized when the molecules are most evenly distributed between the boxes (meaning that their pressures are equal). Is it really so "disordered" to say that all the molecules are neatly arranged into equal numbers in each box? Maybe not in a literary world, but it is the only logical conclusion of a proper definition of "entropy." It is not necessary to watch the videos on probability before watching this one, but it may help. And it might help to re-watch the probability videos after watching this one. Moreover, you might like to see how the numbers relate to the equations through sample calculations (uakron, 15min). These sample calculations show how to compute the number of microstates and probabilities given particles in Box A, B, etc and also the change in entropy.
Comprehension Questions:
1. What is the number of total possible microstates for: (a) 2 particles in 2 boxes (b) 5 particles in 2 boxes (c) 10 particles in 3 boxes.
2. What is the probability of observing: (a) 2 particles in Box A and 3 particles in Box B? (b) 6 particles in Box A and 4 particles in Box B? (c) 6 particles in Box A and 4 particles in Box B and 5 particles in Box C?
Relating the microscopic perspective on entropy to macroscopic changes in volume (uakron.edu, 11min) Through the introduction of Stirling's approximation, we arrive at a remarkably simple conclusion for changes in entropy relative to the configurations of ideal gas molecules at constant temperature: ΔS = Rln(V2/V1). This makes it easy to compute changes in entropy for ideal gases, even for subtle changes like mixing.
Comprehension Questions:
1. Estimate ln(255!).
2. A system goes from 6 particles in Box A and 4 particles in Box B to 5 particles in each. Estimate the change in S(J/K).
3. A system goes from 6 moles in Box A and 4 moles in Box B to 5 moles in each. Estimate the change in S(J/mol-K).
Molecular Nature of S: Thermal Entropy (uakron.edu, 20min) We can explain configurational entropy by studying particles in boxes, but only at constant temperature. How does the entropy change if we change the temperature? Why should it change if we change the temperature? The key is to recognize that energy is quantized, as best exemplified in the Einstein Solid model. We learned in Chapter 1 that energy increases when temperature increases. If we have a constant number of particles confined to lattice locations, then the only way for the energy to increase is if some of the molecules are in higher energy states. These "higher energy states" correspond to faster (higher frequency) vibrations that stretch the bonds (Hookean springs) to larger amplitudes. We can count the number of molecules in each energy state similar to the way we counted the number of molecules in boxes. Then we supplement the formula for configurational entropy changes to arrive at the following simple relation for all changes in entropy for ideal gases: ΔS = Cv ln(T2/T1) + R ln(V2/V1). Note that we have related the entropy to changes in state variables. This observation has two significant implications: (1) entropy must also be a state function (2) we can characterize the entropy by specifying any two variables. For example, substituting V = RT/P into the above equation leads to: ΔS = Cp ln(T2/T1) - R ln(P2/P1).
Comprehension Questions:
1. Show the steps required to derive ΔS = Cp ln(T2/T1) - R ln(P2/P1) from ΔS = Cv ln(T2/T1) + R ln(V2/V1).
2. We derived a memorable equation for adiabatic, reversible, ideal gases in Chapter 2. Hopefully, you have memorized it by now! Apply this formula to compute the change in entropy for adiabatic, reversible, ideal gases as they go through any change in temperature and pressure.
3. Make a table enumerating all the possibilities for 3 oscillators with 4 units of energy.
4. Compute the change in entropy (J/k) for 100 oscillators going from 3 units of energy to 50 units of energy.
5. Compute the change in entropy (J/K) for 100 particles going from 3 boxes to 50 boxes. (This is a review of configurational entropy.)
Heat and entropy in a glass of water (uakron, 9min) Taking a glass from the refrigerator causes heat to flow from the room to the water. The temperature of the water slowly rises while the temperature of the (relatively large) room remains fairly constant. Applying the macroscopic definition of entropy makes it easy to compute the entropy changes, but is one larger than the other? Are all entropy changes greater than zero? What does the second law mean exactly?
Comprehension Questions:
1. Describe your own example of a process with an entropy decrease and explain why it doesn't violate the second law.
Molecular Nature of S: Micro to Macro of Thermal Entropy (uakron.edu, 20min) We can explain configurational entropy by studying particles in boxes, but only at constant temperature. How does the entropy change if we change the temperature? Why should it change if we change the temperature? The key is to recognize that energy is quantized, as best exemplified in the Einstein Solid model. We learned in Chapter 1 that energy increases when temperature increases. If we have a constant number of particles confined to lattice locations, then the only way for the energy to increase is if some of the molecules are in higher energy states. These "higher energy states" correspond to faster (higher frequency) vibrations that stretch the bonds (Hookean springs) to larger amplitudes. We can count the number of molecules in each energy state similar to the way we counted the number of molecules in boxes. Then we supplement the formula for configurational entropy changes to arrive at the following simple relation for all changes in entropy for ideal gases: ΔS = Cv ln(T2/T1) + R ln(V2/V1). Note that we have related the entropy to changes in state variables. This observation has two significant implications: (1) entropy must also be a state function (2) we can characterize the entropy by specifying any two variables. For example, substituting V = RT/P into the above equation leads to: ΔS = Cp ln(T2/T1) - R ln(P2/P1).
Comprehension Questions:
1. Show the steps required to derive ΔS = Cp ln(T2/T1) - R ln(P2/P1) from ΔS = Cv ln(T2/T1) + R ln(V2/V1).
2. We derived a memorable equation for adiabatic, reversible, ideal gases in Chapter 2. Hopefully, you have memorized it by now! Apply this formula to compute the change in entropy for adiabatic, reversible, ideal gases as they go through any change in temperature and pressure.
You might better understand the macroscopic definition of entropy (uakron, 9min) if you consider isothermal reversible expansion of an ideal gas. Note the word "isothermal" is different from "adiabatic." If the expansion was an adiabatic and reversible expansion of an ideal gas, then we know from Chapter 2 that the temperature would go down, ie. T2/T1=(P2/P1)^(R/Cp)=(V1/V2)^(R/Cv). Therefore, holding the temperature constant must require the addition of heat. We can calculate the change in entropy for this isothermal process from the microscopic balance, then show that the amount of heat added is exactly equal to the change in entropy (of this reversible process) times the (isothermal) temperature. Studying the energy and entropy balance for the irreversible process helps us to appreciate how entropy is a state function. As suggested by the hint at the end of this video, you can turn this perspective around and infer the relation of entropy to volume by starting with the macroscopic definition and calculating exactly how much heat must be added after adiabatic, reversible expansion in order to recover the original (isothermal) temperature. Through this thought process, you should start to appreciate that the micro and macro definitions are really interchangeable expressions of the same quantity.
Comprehension Questions: (Hint: entropy is a state function.)
1. Use the macroscopic definition of entropy to compute the change in entropy (J/mol-K) of N2 in a piston/cylinder from 450K and 1cm3/mol to 450K and 4cm3/mol.
2. Use the macroscopic definition of entropy to compute the change in entropy (J/mol-K) of N2 in a piston/cylinder from 450K and 4cm3/mol to 258.46K and 4cm3/mol.
3. Use the macroscopic definition of entropy to compute the change in entropy (J/mol-K) of N2 in a piston/cylinder from 450K and 1cm3/mol to 258.46K and 4cm3/mol.
4. Use the macroscopic definition of entropy to compute the change in entropy (J/mol-K) of N2 in a piston/cylinder from 450K and 1cm3/mol to 300K and 3 cm3/mol.
Once we establish equations relating macroscopic properties to entropy changes, it becomes straightforward to compute entropy changes for all sorts of situations. To begin, we can compute entropy changes of ideal gases (learncheme, 3 min). Entropy change calculations may also take a more subtle form in evaluating reversibility (learncheme, 3min).
Comprehension Questions:
1. Nitrogen at 298K and 2 bars is adiabatically compressed to 375K and 5 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
2. Nitrogen at 350K and 2 bars is adiabatically compressed to 575K and 15 bars in a piston/cylinder. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
3. Steam at 450K and 2 bars is adiabatically compressed to 575K and 15 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
4. Steam at 450K and 2 bars is isothermally compressed to 8 bars in a continuous process. (a) Compute the entropy change. (b) Is this process reversible, irreversible, or impossible?
Simplifying the complete entropy balance (uakron, 11min) is analogous to simplifying the complete energy balance. In general, there are fewer terms to worry about because the system's kinetic and gravitational energy are not involved. This presentation focuses on the same three most common systems as for energy applications.
Comprehension Questions. Write the simplified entropy balance for the following:
1. An isothermal reversible compressor.
2. The expansion stroke of an 80% efficient piston/cylinder.
3. An adiabatic, reversible, ideal gas expanding in a piston/cylinder.
4. Steam expanding through an adiabatic, reversible turbine.
General procedure to solve for steam turbine efficiency. (LearnChemE.com, 5min) This video outlines the procedure without actually solving any specific problem. It shows how inefficiency affects the T-S diagram and how to compute the actual temperature at the turbine outlet.
Comprehension Questions:
1. In this video, the entropy at the outlet of the actual turbine is to the right of the entropy for the reversible turbine. Suppose we were interested in the T-S diagram for a 75% efficient compressor. Would the outlet entropy of the actual compressor be to the right of the entropy for the reversible turbine, to the left, or about the same? Explain.
2. In the video, Prof. Falconer states that the outlet entropy must be the same as the inlet entropy because the process is reversible and one other property. What is the other requirement for the turbine to be isentropic? Explain.
3. Will inefficiency in the turbine always cause the temperature at the outlet to be higher than the inlet? Explain.
Entropy Balances: Solving for Turbine Efficiency Sample Calculation. (uakron.edu, 10min) Steam turbines are very common in power generation cycles. Knowing how to compute the actual work, reversible work, and compare them is an elementary part of any engineering thermodynamics course.
Comprehension Questions:
1. An adiabatic turbine is supplied with steam at 2.0 MPa and 600°C and it exhausts at 98% quality and 24°C. (a) Compute the work output per kg of steam.(b) Compute the efficiency of the turbine.
2. A Rankine cycle operates on steam exiting the boiler at 7 MPa and 550°C and expanding to 60°C and 98% quality. Compute the efficiency of the turbine.
Turbine calculations using Steam.xlsx (uakron.edu, 15min) Using the Steam.xlsx spreadsheet can facilitate computations by eliminating the need for interpolation. You may have seen this video before, but it is convenient to link it here too since turbines often operate on steam.
Comprehension Questions: Solve the following using Steam.xlsx for the vapor properties.
1. An adiabatic turbine is supplied with steam at 2.0 MPa and 600°C and it exhausts at 98% quality and 24°C. (a) Compute the work output per kg of steam.(b) Compute the efficiency of the turbine.
2. A Rankine cycle operates on steam exiting the boiler at 7 MPa and 550°C and expanding to 60°C and 98% quality. Compute the efficiency of the turbine.
Compressor efficiency using an ideal gas assumption (uakron.edu, 13min) Propane is compressed from -100F and 1 bar to 180F and 10 bar. This is enough information to compute the efficiency of the compressor. In this video, we use the ideal gas assumption. We solve the same problem later using more accurate property estimates. Re-watching this video after you have solved the problem using the chart will help you to understand a lot about the influences of molecular interactions and their significance in accounting for the work that goes into designing a chemical engineering process.
Comprehension Questions:
1. An ordinary vapor compression cycle (OVC) is to be considered for cryogenic cooling. The process fluid is to be propane with a compression/expansion ratio (ie. PHi/PLo) of 5.2. The evaporator coils operate at 0.148MPa. The adiabatic compressor's actual exit temperature is 120°F. You may assume the ideal gas law. Hint: what temperature is implied by the pressure of 0.148MPa for the "evaporator." (cf. Eqn. 2.47).
(a) Write the energy balance for the compressor.
(b) Estimate the actual work required for this compressor.
(c) Write the entropy balance required to estimate the efficiency of the compressor.
(d) Estimate the reversible work required for this compressor.
(e) Estimate the compressor's efficiency.
2. HFC134a is to be considered as the working fluid in a prospective refrigeration system. HFC134a (MW=102) is compressed in an adiabatic compressor from 244K and saturated vapor to 316K and 0.9856MPa. Assume the ideal gas law.
(a) Write the energy balance for the compressor.
(b) Estimate the actual work required for this compressor.
(c) Write the entropy balance required to estimate the efficiency of the compressor.
(d) Estimate the reversible work required for this compressor.
(e) Estimate the compressor's efficiency.
1. An ordinary vapor compression cycle (OVC) is to be considered for cryogenic cooling. The process fluid is to be propane with a compression/expansion ratio (ie. PHi/PLo) of 5.2. The evaporator coils operate at 0.148MPa. The adiabatic compressor's actual exit temperature is 120°F. Whenever using the chart, show your work on the attached chart.27%
a. Write the energy balance for the compressor. (3)
b. Estimate the required properties at the compressor inlet to estimate the work.(3)
c. Estimate the required properties at the compressor outlet to estimate the work.(3)
d. Estimate the actual work required for this compressor. (3)
e. Estimate the coefficient of performance of a Carnot cycle operating between equivalent inlet and outlet conditions.(5)
f. Write the entropy balance required to estimate the efficiency of the compressor.(3)
g. Estimate the required properties to estimate the efficiency of the compressor.(4)
h. Estimate the compressor's efficiency.(6)
How to read a pressure-enthalpy chart (uakron.edu, 9min) In principle, reading properties from a chart is no different from looking them up in a table (like the steam tables). In some ways, you could argue that it is easier because interpolation is unnecessary. On the other hand, there are so many lines of the propane chart, all going in different directions, it can be a little confusing at first. In general, the best approach is to use the saturation table when you can, and read the chart when necessary. This video walks you through the process.
Comprehension Questions:
1. HFC134a is to be considered as the working fluid in a prospective refrigeration system. HFC134a (MW=102) is compressed in an adiabatic compressor from 244K and saturated vapor to 316K and 0.9856MPa. (a) Estimate the pressure(MPa), enthalpy (J/g) and entropy(J/g-K) for the compressor inlet. (b) Estimate the enthalpy (J/g) and entropy(J/g-K) for the compressor outlet.
Compressor efficiency using real propane (uakron.edu, 11min) Propane is compressed from -100F and 1 bar to 180F and 10 bar. This time we solve for the compressor efficiency using the chart to estimate the thermodynamic properties.
Comprehension Questions:
1. Re-watch the video showing the solution of this problem based on the ideal gas law. What is the temperature exiting an adiabatic, reversible compressor assuming the propane inlet above? How does that compare to the temperature for an adiabatic, reversible ideal gas? Explain why one is higher than the other.
2. HFC134a is to be considered as the working fluid in a prospective refrigeration system. HFC134a (MW=102) is compressed in an adiabatic compressor from 244K and saturated vapor to 316K and 0.9856MPa. (a) Write the relevant energy balance. (b) Write the relevant energy balance. (c) Solve for the actual work (J/g) (d) Estimate the efficiency of the compressor.
Isothermal compression of steam (uakron, 11min) Compute the work of isothermally and reversibly compressing steam from 5 bars and 224°C to 25 bars. Pay close attention to the problem statement!
Comprehension Questions:
1. Two moles of methane at 3bar and 200K are isothermally and reversibly compressed to 30 bar in a piston/cylinder.
Assume the ideal gas law.
(a) Write the energy and entropy balances.
(b) Estimate the change in entropy (J/K) and enthalpy (J).
(c) Solve for the work(J/g).
2. Two moles of methane at 3bar and 200K are isothermally and reversibly compressed to 30 bar in a piston/cylinder. Use the chart and table for methane.
(a) Write the energy and entropy balances.
(b) Estimate the change in entropy (J/K) and enthalpy (J).
(c) Solve for the work(J/g).
(d) Compare the changes in entropy and enthalpy for real methane to those for ideal gas methane.
Using the NIST WebBook for the propane compression problem (uakron, 14min). The NIST WebBook makes it just as easy to solve problems for propane (and 50 other fluids) as it is for steam. They effectively provide "steam" tables for 50 fluids besides steam.
Comprehension Questions:
1. Re-solve the R134a problem above using the WebBook.
2. Re-solve the methane problem above using the WebBook.