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Understanding Thermodynamics (Dover Books on Physics) by H.C.Van Ness (1-Sep-1983) Paperback
Clearly written treament elucidates fundamental concepts and demonstrates their plausibility and usefulness. Language is informal, examples are vivid and lively, and the perspectivie is fresh. Based on lectures delivered to engineering students, this work will also be valued by scientists, engineers, technicians, businessmen, anyone facing energy challenges of the future.
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First published January 1, 1983
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Displaying 1 - 27 of 27 reviews
September 7, 2016
Very good book for doing a casual walk-through of the first and second law thermodynamics. The restating of Feynman's example for first law of thermodynamics was just enjoyable. The Carnot engine discussion was very enjoyable and very good.
September 29, 2014
A nifty little book which is a good companion to a more rigorous treatment. His style is very Feynmanesque and lucid. The cost of this approach is that some subjects appear out of nowhere (ideal gas internal energy depends solely on temperature during his discussion on cycles) and he is fast and loose philosophically (inexact differentials).
July 26, 2022
This book is really helpful little primer on thermodynamics. The approach of van Ness is much better than a lot of the standard ones I have seen so far. Normally authors can't wait to introduce the big concepts like all the laws right out of the gate. Van Ness makes a point of extending the Feynman energy story at the start.
What is really interesting at the start here is how he discusses what energy is. He pretty much says that it is a mathematical construct, a function of other real variables, and thus isn't something that we can actually find or know in the real world. It is a property of a system, related to the volume, temperature, pressure, and number of particles and it happens to be conserved. That is the extent of our knowledge.
Van Ness introduces the idea of reversibility very well. Reversible processes are the processes that occur through slow, differential steps that can proceed in either direction. The differential steps are essential to allowing us to calculate and the reversible process is the best we can imagine. Van Ness discusses heat engines and heat pumps and their relation to reversibility as well as a very good example involving a piston and infinitesimal masses. Reversible processes are those that extract the maximum work or require the minimum work in expansion or compression respectively.
When getting around to the second law van Ness is careful not to just claim that entropy is not just a measure of disorder. In fact, he describes it in perhaps the best way I have come across yet. First, he starts by telling us about the different relationships that we could investigate between thermodynamic quantities P, W, V, T and Q. If we look at the relationship between 1/P and W we find that all paths from 1/P_1 to 1/P_2 have equal areas under their curves. In other words ∫1/P dW from 0 to W_final is the same for all paths. This relationship must mean something and when we take a step back we realize that this is the definition of the volume. Without the possibility of simply measuring that quantity the only thing we would have to give us insight into the volume is this relationship. What about other possible relations?
Going through a few more we find another example of a relationship between two variables that has the same property of having equal areas under the curves from point 1 to point 2. This time the relation is ∫1/T dQ = ∆S. This is the entropy and van Ness says that we know as much about it as we do energy. Really the agreement about entropy extends about as far as its name, but this is likely only true in this dated account. So there is some relationship between the heat and the temperature, or we could think about it as the heat and the internal energy of a noninteracting system of particles, that we can describe as entropy. This is much the same as the relationship between several variables and the quantity we call energy. With energy there is a basic rule as determined by, or maybe more in line with van Ness as is described by, the first law. It simply states the ∆E=0. For entropy we need to investigate things to see what exactly is going on.
Looking again at a piston in a container with gas we know that the reversible process would extract the most work out of the expansion. This means that an irreversible process would do the same while lowering the energy inside the system less, since it produces less work. Since the internal energy is decreased by a lower amount in the irreversible process if we want to get to the same temperature point as we would if we did a reversible expansion we must remove energy from the system in the form of heat to reach that point. The irreversible process itself is adiabatic, therefor not causing the increase in entropy. In real life processes aren't perfectly adiabatic, and they don't occur in differential steps, so all processes that occur on macro scales involving lots of parts that can be described by thermodynamics are irreversible and not adiabatic. So in this way all of these processes end up losing heat and thereby increasing their entropy. It is true for expansion of gases, where irreversible processes give less than maximum work extracted, and it is true for compression processes which end up requiring more energy than we expect from the reversible process. The second law can be summed up as ∆S≥0.
The rest of the book is dedicated to exploring the second law a bit more. This makes sense. Entropy is very abstract and confusing, a point which I understand from my own experience and one that van Ness details competently. He makes it clear that the summary just above really applies to total entropy, so ∆S_total = ∆S_surroundings + ∆S_system ≥ 0. This is significant because it of course means that a system itself may be brought to a lower entropy state as long as the entropy of the entropy of the surroundings increase. The other interesting point is that he makes the argument that if a system is isolated, and the internal conditions are in equilibrium, that means that the entropy would have maximized, simply because if not then that would mean processes would still be occurring, and they would do so in such a way that would lead to entropy increasing. Or if they were occurring in any meaningful way within the system they would do so in a way that doesn't increase entropy, thereby meaning entropy would trend towards a maximum at equilibrium.
The final chapter is very interesting because it focuses on the statistical mechanics side of things. Van Ness writes about Maxwell's demon for a little while and then goes into some combinatorial mathematics and basic quantum energy levels to derive the Boltzmann distribution in a way that is also much nicer than any I have seen. The Boltzmann distribution is so pervasive in stat mech because it is the distribution of particles along the various quantized energy states that maximizes entropy. When the system has reached equilibrium, as we saw before, it will have evolved towards maximum entropy.
I have always had a gripe with thermodynamics and statistical mechanics. The biggest issue is philosophical. A large part of me wants to be describing the world as it actually is when I am doing physics. In non-statistical physics like theoretical mechanics, there are points where we make assumptions that I don't really like because it feels kind of cheap to me. It almost feels like we are admitting defeat and saying we'll take what we can get. Which is really what we have to do. When we are dealing with coupled oscillators and making the small angle approximation it does simplify the math, but I always feel dirty knowing I am still thinking about big angles. It is something I can normally get past with other branches. With thermodynamics and statistical mechanics this squinting and just taking what we can get comes in higher doses. We look at the whole collection of particles and find what they probably look like and how they interact or behave on average. It all works out very well in terms of predictive power. Van Ness summarizes my issue pretty well when he said that "thermodynamics merely puts a limit on genius." Part of me is reluctant to accept this limit.
Another (or maybe the same) part of me feels like there is always some philosophical sleight of hand at work when entropy ends up getting used to talk about order and information. Van Ness to his credit, again, does much better here than anyone else I have read thus far. He describes Maxwell's demon working on separating particles by some rule. This rule is dependent on information about the particles, whether that be separating the fast and slow particles, the different species, or even the things that are particles from the nothing. In order to do this there needs to be some amount of information about the particles as they are approaching the demon's partition and this can't occur mechanically (at least Van Ness makes a convincing enough reason to believe so, though I could always be swayed). If there were mechanical interaction by a particle and something more massive than it, than the motion of the particle would be disrupted and sorting couldn't occur. If the sorting mechanism was on the scale of the particles it would be subject to the same thermal motion, and would thus not be a stable source of information gathering. Thus we must have information move through some sort of electromagnetic interactions. Van Ness says only these sorts of interactions can conduct information and part of this is confusing, since all mechanical interactions are actually electromagnetic interactions. I think I get his point here though.
So, I can kind of understand his point about information being related to ordering. The biggest issue I have with this book comes in the leap from ordering to entropy (not surprising). Van Ness sort of just says that when we go about ordering things by use of the demon, that is using information and the process itself happens to lower entropy. I think the introduction of the demon always makes things unnecessarily involved. It is entirely possible that even without the demon a chamber of gas connected to another chamber of gas by a small tube will spontaneously have of the particles move back to the original side on their own again. This process, by the order/statistical definition of entropy is a lowering of entropy, can occur simply by allowing mechanical/kinetic features of the particles play out. No demon required. Of course, it is very unlikely, but not impossible. This is pointed out by Van Ness and is part of the reason I don't like the second law, at least not the statistical mechanics version of it. It happens to be statistically true that over time entropy increases or stays the same, and as systems get more and more moving parts it becomes forever more likely to stay that way forever, but when a law breaks down to a box with a few free billiard balls bouncing around inside I don't think it really gets to be a law.
All in all though, Van Ness has done amazing work here. He admits shortcomings in our knowledge honestly and makes sure to draw attention to them. This is the spoon full of sugar I think I need when dealing with TD/SM. He explains things in the best order I have seen, and does so in a patient way, first building his way up from strong foundations. This makes the most convincing argument for TD as it stands. Where we have something that just agrees with the data (which is the goal for a large number of physicists, even though it kind of makes my skin crawl (I want to be able to explain the world, not just describe it. That is probably my problem)) Van Ness will say so and that the equation is true because it works. Clear, concise, almost always insightful, this is currently my favorite book on the topic.
What is really interesting at the start here is how he discusses what energy is. He pretty much says that it is a mathematical construct, a function of other real variables, and thus isn't something that we can actually find or know in the real world. It is a property of a system, related to the volume, temperature, pressure, and number of particles and it happens to be conserved. That is the extent of our knowledge.
Van Ness introduces the idea of reversibility very well. Reversible processes are the processes that occur through slow, differential steps that can proceed in either direction. The differential steps are essential to allowing us to calculate and the reversible process is the best we can imagine. Van Ness discusses heat engines and heat pumps and their relation to reversibility as well as a very good example involving a piston and infinitesimal masses. Reversible processes are those that extract the maximum work or require the minimum work in expansion or compression respectively.
When getting around to the second law van Ness is careful not to just claim that entropy is not just a measure of disorder. In fact, he describes it in perhaps the best way I have come across yet. First, he starts by telling us about the different relationships that we could investigate between thermodynamic quantities P, W, V, T and Q. If we look at the relationship between 1/P and W we find that all paths from 1/P_1 to 1/P_2 have equal areas under their curves. In other words ∫1/P dW from 0 to W_final is the same for all paths. This relationship must mean something and when we take a step back we realize that this is the definition of the volume. Without the possibility of simply measuring that quantity the only thing we would have to give us insight into the volume is this relationship. What about other possible relations?
Going through a few more we find another example of a relationship between two variables that has the same property of having equal areas under the curves from point 1 to point 2. This time the relation is ∫1/T dQ = ∆S. This is the entropy and van Ness says that we know as much about it as we do energy. Really the agreement about entropy extends about as far as its name, but this is likely only true in this dated account. So there is some relationship between the heat and the temperature, or we could think about it as the heat and the internal energy of a noninteracting system of particles, that we can describe as entropy. This is much the same as the relationship between several variables and the quantity we call energy. With energy there is a basic rule as determined by, or maybe more in line with van Ness as is described by, the first law. It simply states the ∆E=0. For entropy we need to investigate things to see what exactly is going on.
Looking again at a piston in a container with gas we know that the reversible process would extract the most work out of the expansion. This means that an irreversible process would do the same while lowering the energy inside the system less, since it produces less work. Since the internal energy is decreased by a lower amount in the irreversible process if we want to get to the same temperature point as we would if we did a reversible expansion we must remove energy from the system in the form of heat to reach that point. The irreversible process itself is adiabatic, therefor not causing the increase in entropy. In real life processes aren't perfectly adiabatic, and they don't occur in differential steps, so all processes that occur on macro scales involving lots of parts that can be described by thermodynamics are irreversible and not adiabatic. So in this way all of these processes end up losing heat and thereby increasing their entropy. It is true for expansion of gases, where irreversible processes give less than maximum work extracted, and it is true for compression processes which end up requiring more energy than we expect from the reversible process. The second law can be summed up as ∆S≥0.
The rest of the book is dedicated to exploring the second law a bit more. This makes sense. Entropy is very abstract and confusing, a point which I understand from my own experience and one that van Ness details competently. He makes it clear that the summary just above really applies to total entropy, so ∆S_total = ∆S_surroundings + ∆S_system ≥ 0. This is significant because it of course means that a system itself may be brought to a lower entropy state as long as the entropy of the entropy of the surroundings increase. The other interesting point is that he makes the argument that if a system is isolated, and the internal conditions are in equilibrium, that means that the entropy would have maximized, simply because if not then that would mean processes would still be occurring, and they would do so in such a way that would lead to entropy increasing. Or if they were occurring in any meaningful way within the system they would do so in a way that doesn't increase entropy, thereby meaning entropy would trend towards a maximum at equilibrium.
The final chapter is very interesting because it focuses on the statistical mechanics side of things. Van Ness writes about Maxwell's demon for a little while and then goes into some combinatorial mathematics and basic quantum energy levels to derive the Boltzmann distribution in a way that is also much nicer than any I have seen. The Boltzmann distribution is so pervasive in stat mech because it is the distribution of particles along the various quantized energy states that maximizes entropy. When the system has reached equilibrium, as we saw before, it will have evolved towards maximum entropy.
I have always had a gripe with thermodynamics and statistical mechanics. The biggest issue is philosophical. A large part of me wants to be describing the world as it actually is when I am doing physics. In non-statistical physics like theoretical mechanics, there are points where we make assumptions that I don't really like because it feels kind of cheap to me. It almost feels like we are admitting defeat and saying we'll take what we can get. Which is really what we have to do. When we are dealing with coupled oscillators and making the small angle approximation it does simplify the math, but I always feel dirty knowing I am still thinking about big angles. It is something I can normally get past with other branches. With thermodynamics and statistical mechanics this squinting and just taking what we can get comes in higher doses. We look at the whole collection of particles and find what they probably look like and how they interact or behave on average. It all works out very well in terms of predictive power. Van Ness summarizes my issue pretty well when he said that "thermodynamics merely puts a limit on genius." Part of me is reluctant to accept this limit.
Another (or maybe the same) part of me feels like there is always some philosophical sleight of hand at work when entropy ends up getting used to talk about order and information. Van Ness to his credit, again, does much better here than anyone else I have read thus far. He describes Maxwell's demon working on separating particles by some rule. This rule is dependent on information about the particles, whether that be separating the fast and slow particles, the different species, or even the things that are particles from the nothing. In order to do this there needs to be some amount of information about the particles as they are approaching the demon's partition and this can't occur mechanically (at least Van Ness makes a convincing enough reason to believe so, though I could always be swayed). If there were mechanical interaction by a particle and something more massive than it, than the motion of the particle would be disrupted and sorting couldn't occur. If the sorting mechanism was on the scale of the particles it would be subject to the same thermal motion, and would thus not be a stable source of information gathering. Thus we must have information move through some sort of electromagnetic interactions. Van Ness says only these sorts of interactions can conduct information and part of this is confusing, since all mechanical interactions are actually electromagnetic interactions. I think I get his point here though.
So, I can kind of understand his point about information being related to ordering. The biggest issue I have with this book comes in the leap from ordering to entropy (not surprising). Van Ness sort of just says that when we go about ordering things by use of the demon, that is using information and the process itself happens to lower entropy. I think the introduction of the demon always makes things unnecessarily involved. It is entirely possible that even without the demon a chamber of gas connected to another chamber of gas by a small tube will spontaneously have of the particles move back to the original side on their own again. This process, by the order/statistical definition of entropy is a lowering of entropy, can occur simply by allowing mechanical/kinetic features of the particles play out. No demon required. Of course, it is very unlikely, but not impossible. This is pointed out by Van Ness and is part of the reason I don't like the second law, at least not the statistical mechanics version of it. It happens to be statistically true that over time entropy increases or stays the same, and as systems get more and more moving parts it becomes forever more likely to stay that way forever, but when a law breaks down to a box with a few free billiard balls bouncing around inside I don't think it really gets to be a law.
All in all though, Van Ness has done amazing work here. He admits shortcomings in our knowledge honestly and makes sure to draw attention to them. This is the spoon full of sugar I think I need when dealing with TD/SM. He explains things in the best order I have seen, and does so in a patient way, first building his way up from strong foundations. This makes the most convincing argument for TD as it stands. Where we have something that just agrees with the data (which is the goal for a large number of physicists, even though it kind of makes my skin crawl (I want to be able to explain the world, not just describe it. That is probably my problem)) Van Ness will say so and that the equation is true because it works. Clear, concise, almost always insightful, this is currently my favorite book on the topic.
October 7, 2021
some of the reviewers here must have read a different book !
there is plenty of maths in it
really it is a text book, although the author does make a bit more effort at helping the reader to understand than does the average uni text book writer of thermo
he gives the famous jelly bean explanation of the first law , borrowed from F as he states
he begins well by explaining no one can tell you what energy is, this come as a shock to many students who then trot out "i can, it is the capacity for doing work"
me "and what is work?"
"work is force times distance!"
me " no, what IS work? work is a form of energy, so you're telling me energy is the capacity for being energy"
as the book progresses there is more maths and less explanation.
if thermo is on your curriculum you may as well give this book a go, it might help
there is plenty of maths in it
really it is a text book, although the author does make a bit more effort at helping the reader to understand than does the average uni text book writer of thermo
he gives the famous jelly bean explanation of the first law , borrowed from F as he states
he begins well by explaining no one can tell you what energy is, this come as a shock to many students who then trot out "i can, it is the capacity for doing work"
me "and what is work?"
"work is force times distance!"
me " no, what IS work? work is a form of energy, so you're telling me energy is the capacity for being energy"
as the book progresses there is more maths and less explanation.
if thermo is on your curriculum you may as well give this book a go, it might help
April 15, 2016
Thermodynamics was never more effortless. Have dreaded the subject throughout the college life. In simple examples and stories, basic concepts of TD are clarified and made easy to recollect.
February 24, 2022
Excellent, clear, concise, insightful. Great "on ramp" for thermal physics self-study. Based on lectures the author did in 1968, he touches on what have become key issues today with respect to power generation and power plants: effects on climate of burning fossil fuel, and thermal pollution. One chapter is dedicated to power plants, the rest is focused on fundamentals -- the presentation is lively and clear, especially the thought model for reversible processes maximizing work done, how the entropy can be used to determine on whether a process will "go". The "box of tricks" example in chapters 5 & 6 and using the second law and entropy calculations to determine if (and under what circumstances) it can work is fabulous.
He skips over some things and assumes them in Chapter 5, and the mathematics heats up in the final chapter (on stat mech) but it's tractable if you have some background.
Great read, highly recommend.
He skips over some things and assumes them in Chapter 5, and the mathematics heats up in the final chapter (on stat mech) but it's tractable if you have some background.
Great read, highly recommend.
February 5, 2019
This book tries to give you an intuitive primer about thermodynamics with minimal use of equations. It can be a good way to start reading about thermodynamics before you jump into textbooks.
It covers the usual topics that come under thermodynamics. Example, the first and the second law of thermodynamics, concept of reversibility, heat engine cycles, power plants etc.
While the intuition this book provided on the subject was useful, I found two shortcomings in this book.
a) this is one book I felt could have gone in to some more detail, to provide a stronger perspective on the subject. I had to spend a lot of time googling to fill in the gaps since the details can be sparse at times.
b) the chapter on statistical mechanics is a nasty surprise, because it completely switches from focusing on providing general perspective on the subject to heavy duty and dry use of equations: the the chapter is in sharp contrast the general style of the rest of the book.
Overall, an average read even though I found it useful before I kick off reading the same subject through resnick and halliday next!
It covers the usual topics that come under thermodynamics. Example, the first and the second law of thermodynamics, concept of reversibility, heat engine cycles, power plants etc.
While the intuition this book provided on the subject was useful, I found two shortcomings in this book.
a) this is one book I felt could have gone in to some more detail, to provide a stronger perspective on the subject. I had to spend a lot of time googling to fill in the gaps since the details can be sparse at times.
b) the chapter on statistical mechanics is a nasty surprise, because it completely switches from focusing on providing general perspective on the subject to heavy duty and dry use of equations: the the chapter is in sharp contrast the general style of the rest of the book.
Overall, an average read even though I found it useful before I kick off reading the same subject through resnick and halliday next!
August 26, 2022
Might not help you pass exams (no exercises and too non-standard) but it provides a different conceptual approach from most other thermodynamic books. Ness points out that one need not begin with the Carnot cycle and that there are equivalent approaches. I believe he cites a text by Zemansky as an example. Cannot find the exact name now because there is no index in this very inexpensive Dover publication. By leaving out so many details which weigh down readers in the standard texts , readers can focus on the key points of thermodynamics. Despite the brevity of the book, only 103 pages, the Boltzmann distribution is proven and Ness even goes through the steps of using the method of Lagrange multipliers in case readers are not familiar with it. It was a useful read for me.
March 31, 2018
A relatively easy to read book intended to develop one's intuition on the study of thermodynamics. Thermodynamics is by no means an easy subject and this book is not meant to be a replacement for a textbook in anyway. It even states that in the beginning of the book and it is evident as it is very much text heavy, trying to describe and explain rather than display much arithmatic. It is meant to supplement the study alongside a much more rigorous text. I found it very helpful, yet I will certainly have to revisit this and other texts to better develop my understanding of the subject.
December 9, 2018
We are in 1983, Hendrick is trying to show thermodynamics from different angles using mundane analogies so students can relate to (say before an university course). He starts explaining conservation of energy (first law) using a "ridiculous" story cooked up by Feynman. You will never forget the first law. NEVER. Because the mother goes incrementally deducing the law using sugar cubes. Her son is playing with the cubes in the living room and she just want to know what happens with them as the time goes by. The boy can eat them. Or throw them in a lake or out of a window so squirrels may eat them. By the end, she figured out the first law of thermodynamics using her own means. This is a powerful methodology because learning becomes very entertaining.
May 16, 2021
Short book, good read. It covers first and second law, focusing the latter on thermodynamic cycles and basic design considerations. I liked the section were a few basic calculations for a nuclear plant are made, specially the permanent temperature increase of the river water, which is used to condense steam.
July 28, 2021
Really accessible treatment. Found this through a stackoverflow recommendation and I’m glad I did. Parts are a little dense for what claims to be a “non-rigorous” book, but overall really approachable. Also the cover art is sick.
June 1, 2019
Helped a lot with my class, not a “fun” read.
February 7, 2021
Some good insights. Very much an engineering perspective.
January 1, 2022
Vague and roundabout
May 5, 2023
My professor recommended this book in class. Does a great job of conceptualizing entropy, energy, the laws of thermodynamics and even some statistical mechanics.
December 10, 2016
A friendly, demystifying look at the laws of thermodynamics. It's good to have someone tell you, up front, that many of the "basics" of science are simply codifying of millions of observations, without a solid idea of what's behind it all. Most of this little book avoids mathematics, for the math trogs (like me) among us, though a chapter or two near the end require a firm math knowledge. It's best for Van Ness's welcoming voice.
February 28, 2017
I think it's a good book for high school students to get familiar with Thermodynamics.
also I suggest Engineering students to read the last chapter of book that it's about statistical mechanics. it's a good intro to the subject.
also I suggest Engineering students to read the last chapter of book that it's about statistical mechanics. it's a good intro to the subject.
July 15, 2022
So good. This should be required reading for anyone making arguments about how electric cars make no sense because the same fuel may be used in the car and the electrical plant.
September 25, 2019
What a beautiful book! It's short, its ideas are universal, it's very inexpensive and portable, it's a deserved classic, and it sports a truthful title. While there are definitely some tricky spots, if you've taken calculus, most of it's smooth as, um, cinderblocks. These are the best explanations of basic thermodynamics I've found anywhere. I enjoyed hearing echoes of Feynman's tellings in Van Ness's descriptions of engines and conserved energy; in my opinion, Van Ness's versions improve on Feynman's. Oh. This next thing is-too a good thing: there's enough math to confuse most people who have taken calculus and physics - somewhere in these pages, when things get technical. For example, I got stuck on the description of the vortex tube used for refrigeration and some detailed accounting of specific heat and British Thermal Units. Nevertheless, most of this book is given to illustrating the ideas in wonderfully lucid normal words, and I believe anyone really curious about thermodynamics can learn from it. After that, look, honestly I barely know what I'm talking about, but you will probably have to find some videos or a textbook and some problems to work. Still, technicalities make a lot more sense when they don't seem random, and that's where this book really glows.
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September 23, 2012I understood about 10% of this book, maybe I'm dumb, reading this sure makes me feel dumb or maybe it's that fact I never even took calucalus in high school or that I can't foucus to well to grasp what is being said. I didn't get very far in this book, but I'm marking it was read, and if that don't sit well with you then I double dog dare you to read this book and explain 11% of it to me. Well anyone?
December 29, 2017
The book was enjoyable and covered the first and second law of Thermodynamics in an easily accessible manner. I especially like the retelling of the sugar-cube story first related by Richard P. Feynman. The book also discusses the Carnot Cycle in an interesting way. The only problems I have with the book are that it is so short and that there aren't any problems to solve. However, the length is such that it is easy to leaf through and find a particular item, so I am rather torn on that.
January 24, 2008
Brilliantly concise.
December 21, 2009
I still don't understand thermodynamics very well...probably because I don't understand the differential equations used in the book. The concepts were interesting though.
September 1, 2013
Good book
December 12, 2024
Good book complemented by an insightful chapter on the link between thermodynamics and statistical mechanics.
Displaying 1 - 27 of 27 reviews