User talk:Paul Wormer/scratchbook: Difference between revisions

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A '''physical system''' is the part of the universe that a physicist is interested in. [[Physics]] is a [[reductionism|reductionist]] science meaning that a physicist restricts his<ref>For linguistic reason we write "he" and "his" when referring to a physicist. This does not imply that physicists are necessarily male.</ref> studies to that part of the universe that is as simple as possible and yet shows&mdash;as far as he can see&mdash;all the physical phenomena that are essential to his study. This delimitation of his object of study is a ''conditio sine qua non''  in understanding and
The '''second law of thermodynamics''', as formulated in the middle of the 19th century by [[William Thomson]] (Lord Kelvin) and [[Rudolf Clausius]], states that it is impossible to gain mechanical energy by letting heat flow from a ''cold'' to a ''warm'' object. The law states, on the contrary, that mechanical energy (work) is needed to transport heat from a low- to a high-temperature heat bath.  
explaining his observations.


Hand in hand with ''reduction'' go ''idealization'' and ''abstraction''. Non-physicists are
If the second law would be invalid, there would be no energy crisis. For example, it would be possible—as already pointed out by Lord Kelvin—to fuel ships by energy extracted from sea water. After all, the oceans contain immense amounts of [[internal energy]]. If it would be possible to extract a small portion of this energy—whereby a slight cooling of the sea water would occur—and  to  use this energy to move the ship (a form of work), then the seas could be sailed without any net consumption of energy.  It would ''not'' violate the [[first law of thermodynamics]], because the  the ship's rotating propellers do heat the water and in total the energy of the supersystem "ship-plus-ocean" would be conserved, in agreement with the first law.  Unfortunately, it is not possible, because a ship is warmer than the sea water that it moves in (or at least not colder) and hence no work can be extracted from the water by the ship.  
amused by the idealizations commonly applied in physics. Many have heard in high
school of the proverbial infinitely thin, infinitely strong, yet massless, rope from which hangs a heavy mass of infinitely small diameter. Many non-physicists are deterred by the abstractions that have entered physics over the last three centuries. What does it mean that a physical system strives for maximum [[entropy]] or that a [[wave function]] of a system collapses when measurements
are performed on it? What exactly vibrates when a radio signal is emitted? It takes
physics students quite some time and effort before they can visualize in their minds these concepts.
Interested laymen are often irritated by the abstractions of physicists that they conceive
as unnecessary ''Wichtigmacherei'' (making important).


When a physicist separates  part of the universe as his physical system, i.e., as his object of
==Entropy==
study, then he must define  at the same time  the variables that determine the ''state'' of the
Clausius was able to give a mathematical expression of the second law. In order to be able do that, he needed the concept of [[entropy]]. Following his footsteps entropy will be introduced in this subsection.  
system. Without the concept of state the concept of physical system is valueless. When
[[Newton]] considered around 1666 his physical system to consist of the point masses [[Sun]]
and [[Earth]], he simultaneously assumed that the state of this system is uniquely
determined by the position and the velocity of the Earth. In this he made the
idealizing assumptions that the Sun is at rest and that the diameters of Sun and Earth are
of no importance and may be set equal to zero. (When Newton later explained the origin of
the tides, the diameter of the Earth became, of course, non-negligible).


Most physical states are non-stationary, they develop in time. The pertinent parameters,
The state of a thermodynamical system is characterized by a number of (dependent) variables, such as [[pressure]] ''p'', [[temperature]] ''T'', amount of substance, volume ''V'', etc. In general a system has a number of energy contacts with its surroundings. For instance, the prototype thermodynamical system, a gas-filled cylinder with piston, can perform work ''DW'' = ''pdV''   on its surroundings, where ''dV'' stands for a small increment of the volume ''V'' of the cylinder, ''p'' is the pressure inside the cylinder and ''DW'' stands for a small amount of work. This small amount is indicated by ''D'', and not by ''d'', because ''DW'' is not necessarily a differential of a function.  However, when we divide by ''p'' the quantity ''DW''/''p'' becomes equal to the differential of the state function ''V''. State functions are local, they dependent on the actual values of the parameters, and not on the path along which the state was reached. Mathematically this means that integration from point 1 to point 2 along path I is equal to integration along another path II
which&mdash;by another idealization&mdash;are assumed to be observable (measurable),
:<math>
change in time. The main purpose of physics is to discover the laws that describe
V_2 =  V_1 + {\int\limits_1\limits^2}_{{\!\!}^{(I)}} dV
the development in time of the state of the physical system. When a physicist sets himself to the task of discovering these laws, he makes the
=  V_1 + {\int\limits_1\limits^2}_{{\!\!}^{(II)}} dV
assumption that the time development of a state is ''causal'', that is to say, that a state at certain
\;\Longrightarrow\; {\int\limits_1\limits^2}_{{\!\!}^{(I)}} \frac{DW}{p} =
time uniquely fixes the state of the same system at a later time. Further he will search for
{\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DW}{p}
the mathematical equation that will describe the time development. This is the ''equation of motion'' of the physical system. Newton discovered by his study of two attracting masses
</math>
his famous second law '''F''' = ''m'' '''a''' and Schrödinger discovered ''H''&Psi;= ''i d&Psi;/dt'' for the causal development of the wave function &Psi; of a system consisting
The amount of work (divided by ''p'') performed along path I is equal to the amount of work (divided by ''p'')  along path II, which proves that the fraction ''DW''/''p'' is a state variable.  
of microscopic quantum particles.


We saw that a physical system does not have to be separated mechanically from the
Absorption of a small amount of heat ''DQ'' is another energy contact of the system with its surroundings.  In a completely analogous manner, the following result  can be shown for ''DQ'' (divided by ''T'')  absorbed by the system along two different paths:
universe. Indeed, it is evident that Newton did not put the Sun and the Earth inside a
vessel with adiabatic walls, in other words, a physical system is not
necessarily physically isolated from his environment. However, in practice it can be very
convenient if it is separated, because it will aid the interpretation and explanation of
measurements when one is assured that certain interactions with the surroundings are not present.


It is usually not easy for an experimentator to separate a physical system from the rest
<div style="text-align: right;" >
of the universe. For instance, a physical chemist studying a system consisting of
<div style="float: left;  margin-left: 35px;" >
molecules will try to observe only the molecules that he is interested in, and will try to
<math>{\int\limits_1\limits^2}_{{\!\!}^{(I)}}\frac{DQ}{T} = {\int\limits_1\limits^2}_{{\!\!}^{(II)}} \frac{DQ}{T}
reduce the number of other molecules. Thus, he needs very thorough purification and/or
</math>
high vacuum. He will also try to shield the molecules from unwanted external fields, such as
</div>
electrostatic, magnetic, and gravitational fields. (The latter field cannot be shielded,
<span id="(1)" style="margin-right: 200px; vertical-align: -40px; ">(1)</span>
but weightless conditions are possible in space stations). For a theoretician, on the
</div>
other hand, the definition of an isolated physical system is trivial, it is just the part of the
<br><br>
universe (matter and fields) that he considers in his equations.
Hence the quantity ''dS'' defined  by
:<math>
dS \;\stackrel{\mathrm{def}}{=}\; \frac{DQ}{T}
</math>
is the differential of a state variable ''S'', the ''entropy'' of the system. Before proving  equation (1) from the second law, it is emphasized that this definition of entropy  only fixes entropy differences:
:<math>
S_2-S_1 \equiv \int_1^2 dS = \int_1^2 \frac{DQ}{T}
</math>
Note further that entropy has the dimension energy per degree temperature (joule per degree kelvin) and recalling the [[first law of thermodynamics]]  (the differential ''dU'' of the  [[internal energy]] satisfies ''dU'' = ''DQ'' + ''DW''), it follows that
:<math>
dU = TdS - pdV.\,
</math>
(For convenience sake  only a single work term was considered here, namely ''DW'' = ''pdV'').  
The internal energy is an extensive quantity, that is, when the system is halved, ''U'' is halved too. The temperature ''T'' is an intensive property, independent of the size of the system. The entropy ''S'', then, is also extensive. In that sense the entropy resembles the volume of the system. An important difference between ''V'' and ''S'' is that the former is a state variable with a concrete  meaning, whereas the latter is introduced by analogy and therefore is not  easily visualized.


The conceptually most important physical system is the ''closed system'', where it assumed that there
===Proof that entropy is a state variable===
is no interaction with the rest of the universe. No energy or matter can flow in or out of
After equation [[#(1)|(1)]] has been proven, the entropy ''S''  has been shown to be a state variable. The standard proof, as given now, is physical and by means of the construct of [[Carnot cycle]]s and is derived from the Clausius/Kelvin formulation of the second law given in the introduction.
a closed system. Obviously, completely closed systems are of no interest to experimental physicists, because no signals will leave such a system and he will not be able to manipulate the system because no signals will enter a closed system either. Thus, in the laboratory, physical systems are always partly open. For a theoretician the idealizing concept of closed system is of great importance and almost always applied, even in studies of open systems. For instance, when a thermodynamicist considers a system that is in temperature equilibrium with its environment (an open system, heat may flow in and out), he will assume it to be in a very large heat bath and  the original system plus the heat bath is then again a closed physical system.
 
==Note==
<references />

Revision as of 10:18, 27 October 2009

The second law of thermodynamics, as formulated in the middle of the 19th century by William Thomson (Lord Kelvin) and Rudolf Clausius, states that it is impossible to gain mechanical energy by letting heat flow from a cold to a warm object. The law states, on the contrary, that mechanical energy (work) is needed to transport heat from a low- to a high-temperature heat bath.

If the second law would be invalid, there would be no energy crisis. For example, it would be possible—as already pointed out by Lord Kelvin—to fuel ships by energy extracted from sea water. After all, the oceans contain immense amounts of internal energy. If it would be possible to extract a small portion of this energy—whereby a slight cooling of the sea water would occur—and to use this energy to move the ship (a form of work), then the seas could be sailed without any net consumption of energy. It would not violate the first law of thermodynamics, because the the ship's rotating propellers do heat the water and in total the energy of the supersystem "ship-plus-ocean" would be conserved, in agreement with the first law. Unfortunately, it is not possible, because a ship is warmer than the sea water that it moves in (or at least not colder) and hence no work can be extracted from the water by the ship.

Entropy

Clausius was able to give a mathematical expression of the second law. In order to be able do that, he needed the concept of entropy. Following his footsteps entropy will be introduced in this subsection.

The state of a thermodynamical system is characterized by a number of (dependent) variables, such as pressure p, temperature T, amount of substance, volume V, etc. In general a system has a number of energy contacts with its surroundings. For instance, the prototype thermodynamical system, a gas-filled cylinder with piston, can perform work DW = pdV on its surroundings, where dV stands for a small increment of the volume V of the cylinder, p is the pressure inside the cylinder and DW stands for a small amount of work. This small amount is indicated by D, and not by d, because DW is not necessarily a differential of a function. However, when we divide by p the quantity DW/p becomes equal to the differential of the state function V. State functions are local, they dependent on the actual values of the parameters, and not on the path along which the state was reached. Mathematically this means that integration from point 1 to point 2 along path I is equal to integration along another path II

The amount of work (divided by p) performed along path I is equal to the amount of work (divided by p) along path II, which proves that the fraction DW/p is a state variable.

Absorption of a small amount of heat DQ is another energy contact of the system with its surroundings. In a completely analogous manner, the following result can be shown for DQ (divided by T) absorbed by the system along two different paths:

(1)



Hence the quantity dS defined by

is the differential of a state variable S, the entropy of the system. Before proving equation (1) from the second law, it is emphasized that this definition of entropy only fixes entropy differences:

Note further that entropy has the dimension energy per degree temperature (joule per degree kelvin) and recalling the first law of thermodynamics (the differential dU of the internal energy satisfies dU = DQ + DW), it follows that

(For convenience sake only a single work term was considered here, namely DW = pdV). The internal energy is an extensive quantity, that is, when the system is halved, U is halved too. The temperature T is an intensive property, independent of the size of the system. The entropy S, then, is also extensive. In that sense the entropy resembles the volume of the system. An important difference between V and S is that the former is a state variable with a concrete meaning, whereas the latter is introduced by analogy and therefore is not easily visualized.

Proof that entropy is a state variable

After equation (1) has been proven, the entropy S has been shown to be a state variable. The standard proof, as given now, is physical and by means of the construct of Carnot cycles and is derived from the Clausius/Kelvin formulation of the second law given in the introduction.

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