Why is the 2nd law of thermodynamics equivalent to conservation of momentum?

[saubhik]

why does heat flows from a high temperature body to a low temperature body?
The above statement can be concluded from applying conservation of momentum to particles of a system containing a high and low temperature bodies.
But in texts, its written that the above statement is a consequence of the 2nd law of thermodynamics.
So is the 2nd law equivalent to momentum conservation principle?
i mean, suppose we did not know the 2nd law, then could we explain the statement using momentum conservation and forget about entropy?

 

[Doc Al]

why does heat flows from a high temperature body to a low temperature body?
The above statement can be concluded from applying conservation of momentum to particles of a system containing a high and low temperature bodies.

How can you draw that conclusion?

 

[saubhik]

suppose you keep a hot body and a relatively cool body in direct contact and take the two bodies as a system. Particles (molecules or ions or atoms) in the hot body have higher energy and thus displays vigorous motion. The particles of the cold body had lower energy and thus their motion is less vigorous. During a collision between the hot and the cold body particles, conserving momentum, we say that velocity of the colder body particles has increased, which means heat transfer has taken place to the colder body.
Mathematically, suppose hotter body particle has mass m1, velocity v1 strikes (elastic) colder body particle with mass m2, velocity v2 (v2<<v1). Suppose the final velocities are v1' and v2'. So,
m1v1 + m2v2 = m1v1' + m2v2'
and from conservation of energy
m1(v1)^2 + m2(v2)^2 = m1(v1')^2 + m2(v2')^2
wherefrom we obtain
v1+v1'=v2+v2' (assuming v1,v1' and v2,v2' are not equal)
now v1>v2 so v1'<v2'.
So finally, cold body particle has higher velocity than hot body particle. Higher velocity means the increase in temperature.

 

[Andrew Mason]

suppose you keep a hot body and a relatively cool body in direct contact and take the two bodies as a system. Particles (molecules or ions or atoms) in the hot body have higher energy and thus displays vigorous motion. The particles of the cold body had lower energy and thus their motion is less vigorous. During a collision between the hot and the cold body particles, conserving momentum, we say that velocity of the colder body particles has increased, which means heat transfer has taken place to the colder body.
Mathematically, suppose hotter body particle has mass m1, velocity v1 strikes (elastic) colder body particle with mass m2, velocity v2 (v2<<v1). Suppose the final velocities are v1' and v2'. So,
m1v1 + m2v2 = m1v1' + m2v2'
and from conservation of energy
m1(v1)^2 + m2(v2)^2 = m1(v1')^2 + m2(v2')^2
wherefrom we obtain
v1+v1'=v2+v2' (assuming v1,v1' and v2,v2' are not equal)
now v1>v2 so v1'<v2'.
So finally, cold body particle has higher velocity than hot body particle. Higher velocity means the increase in temperature.

I don't think you can say that in every collision between a molecule in the hot body and a molecule in the colder one, energy will be transferred from the molecule in the hot body to the molecule in the colder one.

What you can say is that, on average, this will occur. So your "proof" will have to look at many collisions and involve a statistical analysis of the transfer of energy over many collisions.

AM

 

[saubhik]

I don't think you can say that in every collision between a molecule in the hot body and a molecule in the colder one, energy will be transferred from the molecule in the hot body to the molecule in the colder one.
AM

you mean that in some cases there will be no change in velocities of either particles. (i.e. v1=v1' and v2=v2'), it will be as though no collision has taken place. This condition also comes from energy and momentum conservation principles. In other cases we will have v2'>v1'.
for the case when v2'>v1' :
For a simpler case, assume m1=m2 i.e. the hot and cold bodies are made of same substance.
so we get v1+v2=v1'+v2' (from momentum conservation) and from both conservation principles we get v1=v2' and v2=v1' i.e the final velocity of the cold body particle equals the initial velocity of the hot body particle and the hot body particle slows down similarly.

When seen over all the collisions between hot and cold body particles and between interparticle collisions (i.e. collisions between 2 hot body particles or between 2 cold body particles) we see that the average energy of the hot body particles decreases and this decrease equals the increase in the average energy of the cold body particles. Thus heat is transferred from hot body to cold body.

Now my question is why do we need entropy at all? OR find a flaw in my reasoning that only introducing the concept of entropy will resolve it.

 

[Andrew Mason]

When seen over all the collisions between hot and cold body particles and between interparticle collisions (i.e. collisions between 2 hot body particles or between 2 cold body particles) we see that the average energy of the hot body particles decreases and this decrease equals the increase in the average energy of the cold body particles. Thus heat is transferred from hot body to cold body.

Now my question is why do we need entropy at all? OR find a flaw in my reasoning that only introducing the concept of entropy will resolve it.

You are attempting to show that energy of high energy particles is transferred to lower energy particles by collisions. Incidentally, you can show that to dissipate the energy of a fast particle in as few collisions as possible, you have it collide with particles of the same mass.

But the problem is that bodies in thermal equilibrium, say a quantity of an ideal gas, do not have all their molecules moving with the same speed. Their speeds vary according to the Maxwell-Boltzmann speed distribution curve. The hot gas, for example, may have molecules that are moving slower than many molecules in the cooler gas. What prevents the faster molecules in the cooler gas colliding with the slower molecules (across an impermeable membrane separating them) and transferring energy from the faster molecules in the cooler gas to the slower molecules in the hotter gas? That would result in heat flowing by itself from the cool to the hot. What law of physics would that violate?

AM

 

[saubhik]

What prevents the faster molecules in the cooler gas colliding with the slower molecules (across an impermeable membrane separating them) and transferring energy from the faster molecules in the cooler gas to the slower molecules in the hotter gas? That would result in heat flowing by itself from the cool to the hot. What law of physics would that violate?

Thanks for your answer.
Doesn't any collision between slower molecules of hotter gas and faster molecules of cooler gas occur? This seems non-intuitive.
How does entropy solve this? I am not getting a "physical picture".

 

[my_wan]

You are attempting to show that energy of high energy particles is transferred to lower energy particles by collisions. Incidentally, you can show that to dissipate the energy of a fast particle in as few collisions as possible, you have it collide with particles of the same mass.

But the problem is that bodies in thermal equilibrium, say a quantity of an ideal gas, do not have all their molecules moving with the same speed. Their speeds vary according to the Maxwell-Boltzmann speed distribution curve. The hot gas, for example, may have molecules that are moving slower than many molecules in the cooler gas. What prevents the faster molecules in the cooler gas colliding with the slower molecules (across an impermeable membrane separating them) and transferring energy from the faster molecules in the cooler gas to the slower molecules in the hotter gas? That would result in heat flowing by itself from the cool to the hot. What law of physics would that violate?

AM

I'm not getting why it would need to break any law. Do we actually need a law that says it is impossible to roll a dice 100 million times and get snakeyes every time? No. We only need to recognize that the probability is so outrageous that observing it in the time since the bib bang is essentially absurd.

In fact what is being implied by requiring some extra law to disallow such extraordinarily unlikely events? Sounds almost to me like the such a law conceived as an extra component, rather than an effective practical derivative component, a lot like the "life force" or God postulates in other conceptions of motive forces.

---
To the original question: Entropy is an extremely useful way of saying you'll never see a fair coin roll heads 100 million times in a row. It is not a law that it can't happen, it is a probability that it will never occur in countless lifetimes. Since the number of slower gas molecules equals the number of faster than average gas molecules the odds alone rules out any sane notion that that you will ever get enough extra collisions between slower than average molecules in the hot and faster molecules in the cold gas to ever witness heat flow from cold to hot.

In the sense that entropy is a law, it a purely probabilistic law which can be violated to varying degrees over the short term, and perhaps to some higher degree given enough countless trillions of years.

 

[my_wan]

Another point of fact. Not only is the 2nd law the result of conservation, rather than a separable law or constraint on the system, it is empirically inferior to the conservation law form of the description as articulated under statistical mechanics. This is exactly what Einstein used Brownian motion to prove, which incidentally was the first proof the atom actually existed.

 

[Andrew Mason]

I'm not getting why it would need to break any law. Do we actually need a law that says it is impossible to roll a dice 100 million times and get snakeyes every time? No. We only need to recognize that the probability is so outrageous that observing it in the time since the bib bang is essentially absurd.

Exactly. The only law it would break would be a statistical law - ie. the second law of thermodynamics.

To the original question: Entropy is an extremely useful way of saying you'll never see a fair coin roll heads 100 million times in a row. It is not a law that it can't happen, it is a probability that it will never occur in countless lifetimes.

It is not countless. If you flip the coin once a second there is only one way to come up heads every time but 2^(10^8)-1 ways it can come up something else. Each 100 million tries takes 100 million seconds, or about 3 years. So it would, on average, turn up 100 million heads in a row roughly 3 x 2^100,000,000 years. The age of the universe is only about 2^34 years. So I think we can safely say that an event with that kind of probability never has and never will happen anywhere in the entire universe.

AM

 

[saubhik]

Thanks for your answers. I think I have got my answer.

 

[Andrew Mason]

In the sense that entropy is a law, it a purely probabilistic law which can be violated to varying degrees over the short term, and perhaps to some higher degree given enough countless trillions of years.

The second law applies to bodies for which a temperature is defined. Such bodies have at least thousands of particles. The probability of the second law being violated for even the smallest such body for any measurable period of time is extremely remote.

AM

 

[my_wan]

Countless = Too lazy to do the math.

Yeah, it's not that difficult to do the math on the odds of a vacuum spontaneously forming around my head also. Too lazy to do that to

I just thought the response made it sound as though the 2nd law was a physical prohibition rather than an intensely unlikely physical event which no physical law strictly forbids. I also think text, even basic text, should include this explanation of the 2nd law, rather than treating is as a priori fundamental.

Entropy remains more generally useful than a classical causal description can replace. Even in classical systems we often do not know the details of the microscopic degrees of freedom available to the system, but it is unnecessary for entropy to remain a viable tool. Nor does in matter how you quantify the degrees of freedom so it's just as useful in a non-classical context. Simply forgetting about entropy in favor of a more descriptive conservation law effect would make a lot of very simple calculations overly complex and require lots of unnecessary information. Nonetheless, it is good to understand that it is not strictly a fundamental law in itself.