Proof

Excerpts

Proof
It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody’s immediate disposal.
Proof
For supposing it were not so, *i.e.* supposing a method could be found by which a process involving generation of heat by friction could be completely reversed, this very method would produce what is identically perpetual motion of the second kind: viz. a change which consists of nothing but the production of work, and the absorption of an equivalent amount of heat.
Proof
As soon as a phenomenon is found to contradict any legitimate conclusions from the second law, this contradiction must arise from an inaccuracy in our first assumption, and the phenomenon could be used for the construction of the above-described engine.
Proof
It increases or decreases according as heat is absorbed or evolved.
Proof
Not a single really rational proof of the second law has thus far been advanced which does not require this fundamental principle, however numerous the attempts in this direction may have been in recent times, nor do I believe that such an attempt will ever meet with success.
Proof
*It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.*
Proof
Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made. It would, it is true, not be equivalent to perpetual motion, for it does not produce work from nothing, but from the heat, which it draws from the reservoir. It would not, therefore, like perpetual motion, contradict the principle of energy, but would, nevertheless, possess for man the essential advantage of perpetual motion, the supply of work without cost; for the inexhaustible supply of heat in the earth, in the atmosphere, and in the sea, would, like the oxygen of the atmosphere, be at everybody’s immediate disposal.
Proof
From the impossibility of perpetual motion of the second kind, it follows, in the first place, that the generation of heat by friction is *irreversible* (*cf.* def. 112).
Proof
The entropy of the gas, therefore, remains constant during the described adiabatic change of state.
Proof
The entropy of a body in a given state, like the internal energy, is completely determined up to an additive constant, whose value depends on the zero state.
Proof
It should, however, be emphasized that equation % [eqn:(53)](53)% is by no means generally true. It holds only in the particular case where the external work performed by the gas is expressed by $p\, dV$.
Proof
We, therefore, put forward the following proposition as being given directly by experience: *It is impossible to construct an engine which will work in a complete cycle, and produce no effect except the raising of a weight and the cooling of a heat-reservoir.* Such an engine could be used simultaneously as a motor and a refrigerator without any waste of energy or material, and would in any case be the most profitable engine ever made.
Proof
*Every physical or chemical process in nature takes place in such a way as to increase the sum of the entropies of all the bodies taking any part in the process. In the limit, *i.e.* for reversible processes, the sum of the entropies remains unchanged.* This is the most general statement of the second law of Thermodynamics.
Proof
It should be emphasized, however, that the form here given is the only one of unrestricted applicability to any finite process, and that no other universal measure of the irreversibility of processes exists than the amount of the increase of the entropy to which they lead.
Proof
It would be absurd to assume that the validity of the second law depends in any way on the skill of the physicist or chemist in observing or experimenting. The gist of the second law has nothing to do with experiment; the law asserts briefly that *there exists in nature a quantity which changes always in the same sense in all natural processes*.
Proof
As the impossibility of perpetual motion of the first kind leads to the first law of Thermodynamics, or the principle of the conservation of energy; so the impossibility of perpetual motion of the second kind has led to the second law, properly designated as the *principle of the increase of the entropy*.

Equations

Proof
-\frac{Q}{\theta}
The entropy change of a heat-reservoir during an infinitely small time element equals minus the heat given to the substance divided by the reservoir temperature.
This equation is in Applications to Homogeneous Systems (Applications to Homogeneous Systems)
Proof
du = c_{v}\, d\theta
For a perfect gas the change of internal energy per unit mass is the specific heat at constant volume times the change of temperature.
This equation is in Molecular Weight (Molecular Weight)
Proof
q = c_{v}\, d\theta + \frac{R}{m} · \frac{\theta}{v}\, dv
For a perfect gas the heat received per unit mass is the sum of the internal-energy change and the work term, written in temperature and volume.
Proof
\phi = c_{v} \log \theta + \frac{R}{m} \log v + \const
The entropy of unit mass of a perfect gas is defined, up to an additive constant, by logarithms of temperature and specific volume.
Proof
\Phi = M\phi = M \left(c_{v} \log \theta + \frac{R}{m} \log v + \const\right)
The entropy of a mass M of a perfect gas is M times the entropy of unit mass.
Proof
d\Phi = M \left(c_{v}\, \frac{d\theta}{\theta} + \frac{R}{m}\, \frac{dv}{v}\right) = \frac{M · q}{\theta} = \frac{Q}{\theta}
On application of heat to a perfect gas the change of entropy equals the absorbed heat divided by the temperature.
Proof
d\Phi = M \left(c_{v}\, \frac{d\theta}{\theta} + \frac{R}{m}\, \frac{dv}{v}\right) = \frac{dU + p\, dV}{\theta}
The change of entropy of a perfect gas equals the sum of internal-energy change and pressure-volume work, divided by temperature, for any process in which temperature and volume change.
This equation is in Applications to Homogeneous Systems (Applications to Homogeneous Systems)
Proof
Q + W = dU
The heat absorbed plus the work done on the substance equals the change of its internal energy (first law).
Proof
\Phi_{1} + \Phi_{2} = \const
In a reversible process of two gases exchanging heat, the sum of their entropies remains constant.
Proof
\Phi_{1} + \Phi_{2} = \Phi_{1}' + \Phi_{2}'\Add{.}
The two-gas system has the same total entropy in its initial and final states.
Proof
\Phi_{1} + \Phi_{2} + \dots + \Phi_{n} = \Phi_{1}' + \Phi_{2}' + \dots + \Phi_{n}'
A system of n gases has the same total entropy in two states, which is the condition for reversible transformation between them.
Proof
\Phi_{1}' + \Phi_{2}' + \dots + \Phi_{n}' < \Phi_{1} + \Phi_{2} + \dots + \Phi_{n}
The supposed final total entropy of the gas system is smaller than its initial total entropy, which leads to a contradiction.
Proof
(\Phi_{1}' + \Phi_{2}' + \dots + \Phi_{n}') - \Phi_{1} - \Phi_{2} - \dots - \Phi_{n-1}\Add{.}
The entropy of the n-th gas is the total final entropy minus the entropies of the first n-1 gases.
Proof
-\tsum \frac{Q}{\theta} \geq 0
The total entropy change of all heat-reservoirs cannot be negative, since no change remains in other bodies.
Proof
\tsum \frac{Q}{\theta} \leq 0
The sum of heat absorbed divided by reservoir temperatures over a cycle is not positive; this is the form in which Clausius first stated the second law.
Proof
W = -p\, dV
When the external pressure equals the pressure of the substance, the work done on it during compression is minus p dV.
Proof
\tsum \frac{Q}{\theta} = 0
For a cyclic process in which each heat-reservoir is at the temperature of the substance, the sum of Q over theta vanishes.
Proof
\tsum \frac{dU + p\, dV}{\theta} = 0
Over a reversible cyclic process of a homogeneous body, the summation of (dU + p dV)/theta vanishes.
Proof
\int_{1}^{2} \frac{dU + p\, dV}{\theta}
The integral of (dU + p dV)/theta from state 1 to state 2 depends only on the two states, not on the path.
Proof
\int_{1\; (\alpha)}^{2} \frac{dU + p\, dV}{\theta} + \int_{2\; (\beta)}^{1} \frac{dU + p\, dV}{\theta} = 0
Over the complete cycle formed by path alpha from 1 to 2 and path beta back to 1, the integral vanishes.
Proof
\int_{1\; (\alpha)}^{2} \frac{dU + p\, dV}{\theta} = \int_{1\; (\beta)}^{2} \frac{dU + p\, dV}{\theta}
The integral from state 1 to state 2 is the same along any two reversible paths between those states.
Proof
\Phi = \int \frac{dU + p\, dV}{\theta}
The entropy of a body in a state is the integral of (dU + p dV)/theta from the zero state, defined up to an additive constant.
Proof
d\Phi = \frac{dU + p\, dV}{\theta}
The differential of entropy equals (dU + p dV) divided by temperature, for any body.
Proof
d\phi = \frac{du + p\, dv}{\theta}\Add{.}
The differential of entropy per unit mass equals (du + p dv) divided by temperature.
Proof
d\Phi = \frac{Q}{\theta}\Add{.}
For a reversible change of volume, the entropy change of a body equals the absorbed heat divided by temperature.
Proof
U = Mu
The energy of a body equals its mass times its energy per unit mass.
Proof
V = Mv
The volume of a body equals its mass times its specific volume.

Problems

No exercises in this chapter.