**J2
****Heat**

**Conversion
of work into heat and of heat into work**

**The first main theorem,
Impossibility of perpetuum mobile**

The first main
theorem of Thermodynamics comprises experienced facts relating to
the *magnitude *of converted energy - conversion of
mechanical work* **into
heat* or of conversion
of *heat *into other forms of energy. It can be
given different forms, for example, as follows: A given amount of
mechanical work yields, converted into heat, always the same
amount of mechanical work. A given magnitude of work is thus
always *equivalent** *to the same amount of heat and
*vice versa*.

Once the units in
which energies are measured have been defined, the relationships
between heat and work or between heat and another form of energy
can be formulated . The units for mechanical energy are *erg* and mkg*, also 10^{7} erg - the *Joule*, the *Watt second*. The unit for heat is the *calorie*.
The number of units of *mechanical** *energy, which are *equivalent* to a calorie, is called the *mechanical heat equivalent**.*

How large is the
amount of heat, generated by 1 mkg* work? One of the earliest measuring
techniques by Joule 1847/1879 employed generated heat to
raise the temperature of quantities of *water*, *measured* in kg, by a *measurable** *number of degrees. The *work** *is obtained directly in mkg*.

Water
is in a cylindrical vessel *A* (Fig. 356);* *in the
water is a shaft *a* with vanes *b*, which are
rotated by a hanging weight *G *as shown. In order that
the vanes do not push the water ahead of them, but work *through it *with as much *friction *as possible, there are radial separating
walls with openings through which the vanes pass with little
margin. The weight *G* falls and turns the shaft. When it
has fallen as far as the string admits, it has performed all of
the work, which it can perform; it can work again, once it has
been rewound to the top. In this way, arbitrarily much work can
be transmitted to the vanes and the water.

The friction
between the vanes and the water generates heat and hence raises
the water's temperature, shown by a thermometer. (Account is
taken of the heating up of the equipment and the outwards loss of
heat during computation of the generated heat.) The falling
weight in kg* times the height of fall in m yields the work spent
on turning the vanes in mkg*. When 427 mkg* work has been
performed, 1 kcal has been generated, that is, the *work** *of 427 kg*, transferred to the
water, caused the same as the input of 1 kcal heat would have
caused, whence 427 mkg* are *equivalent* to 1 kcal.

This number, which determines the work in terms of mkg*
equivalent to 1 kcal of heat, is called the *mechanical equivalent of heat*; its has been determined over and over again by many
methods (Fig. 357). For example, Joule determined it also in 1843
from the work required to compress air in a container to a
certain pressure and from the accompanying increase in
temperature of the air; Mayer 1842 found it from the expansion of a heated gas: His was the *first method used for calculations*; Quintus Icilius 1824-1885 1857 employed the heat of an electric
current. Whatever approach was used, people obtained always
approximately the same value. In 1935, the most *probable value *was 426.9 mkg*** **(Jaeger and Steinwehr,
"Electric Heating", 1915).

Since 1
kcal = 426.9 mkg* and 1 mkg* = 9.81·10^{7}
erg = 9.81 Joule, then 1 cal = 4.186·10^{7} erg = 4.185 Joule
and, moreover, 10^{7} erg = 0.239. We found earlier for the gas constant the value *R *=
8.313·10^{7} erg/degree per degree. If we replace the
mechanical measure of work by the measure of heat, we find

*R* =
8.313·0.239cal/degree = 1.98 cal/degree.

Everybody knows that heat can generate work,even if it is only striking a match on a rough surface, during which the igniting part of the match reaches its inflammation temperature. During every mechanical work like turning, drilling, filing, the machined and machining parts heat up. Every moving part of a machine gets warm in its bearings and at other locations where friction occurs; heat, generated in such a manner, is useless work, whence you try to limit it by greasing moving contacting surfaces.

It is also easily verified
that one can convert work *into
heat*. Moreover, the* **disappearance** *of mechanical work as it generates heat is
demonstrated, for example, by the fall of the weight in Joule's
experiments. The weight, which has fallen to its lowest position,
has no more of work stored. However,
what about *conversion *of heat into work? The steam engine proves that heat can *perform**
*work. But does it *disappear*
in the process? The engineer Hirn of Kolmar
1815-1890 1865 has shown that steam, which flows directly into
the condenser of a steam engine, yields *more *heat
than if it moves, to start with under otherwise equal conditions,
the piston of the steam engine, that is, has performed work.
Moreover, he has measured with a *test person *in a
completely closed room the quantity of oxygen used and the heat
generated. At perfect rest, this person generated 5.2 kcal per
gram of used oxygen. If it pedalled a bicycle like contraption,
that is, performed work which consumed one part of the oxidation
heat, this number decreased to 2 kcal.

The equivalence of heat and work
was first claimed by the Heilbronn doctor Mayer in 1842. He had observed in Java (Indonesia, during
blood letting) that venous blood has there a light red colour,
while, due to a considerable content of carbolic acid, it is dark
red in middle latitudes. It was known that animal heat (like that
of combustion) arises during the oxidation of nourishment and in
the process carbonic acid is generated. Starting from this
insight, Mayer gave his observation the explanation: In order to
maintain its temperature at a constant level in a hot climate
where less heat is lost, the human body demands *less *heat,
whence it develops less carbonic acid. Mayer also
suggested that the *heat
of the oxydization* of nourishment is *converted into mechanical energy**. Independently of Mayer, Helmholtz 1847 has
proposed equivalence of heat and work in a quite general form and
formulated it mathematically in his paper: "*About the conservation of force**"**.
*The equivalence of heat and work, which is present
everywhere in Nature independently of human insights and
particular forms of work, proves their *internal relationship* and justifies the concept, that both of them are only
different *forms of the
same** *something - *energy*.

* This is
the reason for evaluating the nourishing value of food in
calories. The *utilisation
*of the calories, supplied to an
organism, depends on the digestion of the food and other
metabolic processes. In general, carbohydrates (sugar, starch),
fats, albuminous bodies are the three main nutritive substances
of nourishment. The conversion heat (heat tone, cf. below) per 1
g of cane sugar is 4.0 kcal, of fat 9.3 kcal, of casein 5.9 kcal.
All food can be a source of energy for the work of the muscles,
but carbo-hydrates are preferred. The daily *quantity of food *(Karl Voit 1831-1908 1875) for an adult of weight 67 kg with medium
working effort (carpenter, bricklayer) is: 118 g protein, 56 g
fat, 500 carbohydrate = 3055 kcal. - The heart uses about 150 of
these 3000 kcal, that is about 5% (Höber, "Treatise of the
Physiology of the Human", 1920). In fact, it performs at 70
pulse beats per minute in 24 hours about 21 000 mkg*, the
equivalent of about 50 kcal. In actual fact, the muscle of the
heart needs about three times as much, since it, like a caloric
machine, can only convert about 1/3 of the consumed energy into
work. The performance of the heart (mkg*.sec) is about 1/300 PS.

Another form of the first main
theorem says: The total content of energy *U *of a body is
uniquely determined by the *state
*of the body. It is the energy, which
for example the body would yield in the form of heat, if one
could reduce its temperature to absolute Zero and make the
pressure zero. Depending on its *state*, a body has another
energy constant, for example, 1 g of *steam** *at 100º
has a different one to 1g of *water* at 100º. However,
the state of a body is not already characterized by its *aggregate*
state (gaseous, fluid. solid); you must know, in addition, its
temperature, pressure and density.

**State of a body. Equation of
state**

The state of a
body - of a homogeneous, isotropic body of given chemical
constitution, at uniform temperature and uniform internal
pressure - is determined by its mass *m*, its volume *v*
and its temperature *t*. In this state, all its other
properties depend on *m, v* and *t*. In particular,
its *pressure* depends, apart from on the temperature *T,
*only on *m*/*v* - its density - in other
words: On *v*/*m*, the volume of its mass* *unit*,
**its specific
volume**. *The
function *p* = *f*(*v,t***) **is
called the *equation of
state* of the
substance. It resolves all questions regarding its behaviour in
the face of arbitrary changes of temperature, volume and
pressure!

The equation of
state for the ideal gas has the simplest form. The *pv*-Law
of Boyle-Mariotte and the Law of Gay Lussac regarding the temperature dependence of *pv* is *pv
= p*_{0}*v*_{0}(1 + *a**t*). It answers all the
questions regarding the behaviour of the ideal gas during changes in volume,
pressure and temperature. When gasses get close to their
liquefaction, they differ more and more from the ideal state and
their equation of state changes. The relevant equation is (*p
+ *a*/v*²)(*v - b*) = *RT*, the
equation of state of van
der Waals 1873, where *p,
v *and* T* denote the pressure, volume and absolute
temperature, *R *the gas constant, *a* and* b *are
constants, which depend on the character of the gas; *b*
is a volume correction, *a* allows for the mutual
attraction of the gas particles. For large *v, *the
equation becomes the equation of state of the ideal gas, for
small *v* and corresponding *t *the equation of
state of drop forming substances. Thus, this equation embraces
the ideal state of a gas and the fluid state as special cases
(also the transition from the one state into the other, that is,
condensation, and also the *critical
*point,above which
condensation becomes impossible).

If you measure
pressure, in fractions of the critical pressure, the volume in
fractions of the critical volume, the temperature in fractions of
the critical temperature, the equation of state assumes for all
substances the same form. It does not involve anything signifying
the *peculiarity *of a substance (a *definite** *substance). Two arbitrarily
selected substances yield identical diagrams of state; you must
only compare pressure, volume and temperature in the stated
manner - one says: in *corresponding
states*. The law of
agreeing states allows (according to an expression due to Zeeman), as far as thermic behaviour is concerned, to see in
every substance a copy of another one, although in a changed
scale. - Hitherto (1935), there does not exist an equation of
state which embraces all three states of aggregate.

**The first main theorem in the
form ****U**_{2}**
- U**_{1}**=Q + A**

The first main
theorem can also be formulated mathematically, if one start from
the energy content of a body in two different states **1.**
and **2.** with the energies *U*_{1}*
*and *U*_{2} (Planck). If *U*_{2
}is the larger value, then *U*_{2 }- *U*_{1
}is the increase of energy in the transition from State **1.
**to State **2.** Its energy can increase by
a body accepting a quantity of *heat**
Q* and simultaneously a quantity *A *of *another** *form of energy, which we will
denote briefly by work; the energy can be mechanical, electrical,
chemical, etc.** **One has then altogether *U*_{2}*
- U*_{1}*=Q + A.*

The state of a body then
determines simultaneously its energy content. For example, if a
given, measured volume of a gas undergoes an arbitrary change of
state - changes in temperature and pressure - and returns in the
end to its initial state, that is, to the initial temperature and
pressure, its final energy content is the same as it was at the
start. Such a process, which returns to the initial state - but
can in between be arbitrary - is called a *cyclic process*. We can now give the first main theorem
the form: If a body passes through a cyclic process, it has in
the end the *same* energy as at the start, energy losses
and gains cancel each other *during* such a process .

*Hence follows the
impossibility of the perpetuum mobile*! In fact, the hope for the perpetuum
mobile rests on the assumption that, *without provision of energy** *(in contradiction to the first
main theorem), one can continuously *gain*
energy by letting a body or a system of bodies pass through
certain cyclic changes and return to the initial state, that is,
form a sequence of cyclic processes. However, as has been said
already, a *gain** *of work in a cyclic process is
impossible. In a cyclic process, State **1** equals
State **2**, whence *U*_{2}* = U*_{1}.
However, then *Q + A = *0, that is, in a cyclic process,
the sum of the accepted heat and the consumed work vanishes.

**Application of the first main
theorem to an ideal gas**

The *change**
*of the energy content of a body is especially clear if the
body is a gas. If it is as a whole at rest, its energy is just
its *internal** *energy *U*. Hence, if
we take it from a state *U*_{1} to another state *U*_{2}
and measure the associated *external** *effects (*Q + A*), the
equation *U*_{2}* - U*_{1}*=Q +
A *indicates the magnitude of the change of energy and
whether it has grown or dropped. The required experimental
conditions are readily met in Joule's set-up of
1845 (Fig. 358). *R* and *E *are two closed metal
containers with mechanically rigid walls, linked by the tube *D*;
they can be connected or separated by a tap. They are fully
immersed in water.* R *contains at first air, compressed
to 20 atm and *E* a vacuum. The experiment is as follows:
You measure accurately the temperature of the water, then open
the tap between the vessels when air will flow from *R* to
*E* until the pressure in both vessels is the same; after
some time, the temperature is measured again. The result: The
temperature of the water has *not changed*, whence the air, compressed to 20 atm has expanded
(without performance of mechanical work) to twice its volume, *without changing noticeably its
temperature**.*
Yet more exact measurements have yielded a minute cooling, but for the socalled *permanent**
*gases the action
is minimal, and it is *yet*
smaller as the gas at
rising temperature or decreasing pressure approaches the state of
an *ideal** *gas. Hence, according to the
equation *U*_{2}* - U*_{1}*=Q +
A*, the result of the experiment is *U*_{2}*
- U*_{1}*= *0 (since there are neither thermal
nor mechanical external effects), whence *U*_{2}*
= U*_{1}*. *The temperature remained constant,
although the volume doubled. In other words: The inner energy of
the *ideal gas* remains constant in spite of a large
change of *volume*, that is, it only depends on its *temperature* and not on its volume.

However, the experimental
set-up of Fig. 358 is not sufficiently sensitive. Any temperature
increase of
the air would have to be *very** *large in order to become noticeable in the presence of large masses of metal and
the large mass of water; according to a later calculation, under
these conditions, it might be only 1/200º (Pollitzer). A much more sensitive set-up was developed by Lord Kelvin and employed in measurements with Joule (Fig. 359): A pump drives uniformly air through a long
tube, the temperature of which is held constant. At a location,
which is especially protected against heat conduction, the tube
is plugged by compressed wool, silk or similar material. Through
its pores flows the air with friction uniformly. In front of the
plug, the pressure is higher than behind it. A very sensitive
thermometer is located immediately behind it (in 1935, it was a
thermo-element or platinum thermometer). Under lower pressure,
the air flows slowly and uniformly (without noticeable kinetic
energy). In the set-up of Fig. 358, a limited mass of air rushes
into the vacuum, in that of Fig. 359, an unlimited flow of air
moves slowly and steadily from the space at higher pressure into
that at lower pressure; by artificial deceleration of the outward
stream, the air transits immediately into the second state and
indicates its temperature *immediately*. The result: Also here, the changes of
temperature of the gases is so small that one can expect for *ideal gases* temperature *constancy*, that is, ultimately one can say: The
inner energy of an ideal gas depends only on its temperature and
not on its volume.

We have only introduced the Joule-Thomson experiment with a view to the
properties of ideal gases; it is of great technical importance
for *real *gases. Two processes are here superimposed: **1.**
external, **2. **internal work:

**1**. Driving the gas through the throttle *demands* work; on the other hand, the gas *performs** *work behind the throttle by
pushing other quantities of gas ahead of it. The difference of
these performances of work is called the *external *work of the gas. It is measured by the
difference of the products *pv* (Pressure times specific
volume) behind and ahead of the throttle; in both products you
substitute in the case of differential small pressure changes (*dp*)
the same temperature.

**2. **Superimposed on the external work is the *internal *one of the gas overcoming molecular
forces of attraction. It is always positive and especially large
at low temperatures and then always larger than the external
work. The *effect* depends in magnitude and sign on the
temperature of the condensed gas: At room and lower temperatures *all gases become colder*, only hydrogen and helium become
warmer; at sufficiently high temperatures, all gases become
warmer, while, on the other hand, hydrogen also cools, if it has
been cooled at the start to - 80ºC. This change in
sign is called *inversion*, the corresponding temperature the *inversion temperature** *of the Joule-Thomson-Effect. Only when the temperature of a
gas lies *below* it, it can in this manner continue to
become cooler. Carl von
Linde 1842-1934 has used
the Joule-Thomson-Effect in his air-liquefaction machine.

**The first main theorem in the
form ****dQ = dU + pdV**

Another form of the first main
theorem is due to Clausius: Introduce into the system an
infinitesimal quantity of heat *dQ *and ask: What does it cause and what happens to
it? It increases the inner
energy *U *of the body by *dU. *But the increase *dU
*is not *equal *to *dQ, *because one part of *dQ*
is used to perform the work connected with the increase in volume
of the body *due* to its gain in heat. The energy
increase is reduced by this work - we assume that *only *volume-work is performed. How large is this volume-work? We assume that the only external force,
which has to be taken into account, is the pressure acting on the
surface, which - as happens especially frequently - acts at all
points perpendicularly to it and equally strongly. In order to
compute the work, required to overcome this force, we only need
consider the change in volume of the body *as a whole* (not its change in individual
directions). Denote the change in volume by *dV*, the
pressure by *p*. The computation then yields - we can only
quote here the result - for the volume work *p*·*dV*,
whence we obtain for the change in inner energy *dU = dQ *-*
pdV* or *dQ *= *dU + dV. *This equation is one
of the most used mathematical forms of the first main theorem. We
obtain it from *U*_{2}* - U*_{1}*=Q
+ A* by assuming that *U*_{2}* - U*_{1}*
*is so small, that we can call their difference *dU*,
moreover* *calling the heat introduced from outside *dQ* and setting *A* = - *pdV*; it is
negative, because the work is performed *by* the system (not as before transferred from outside *to* the
system). It will not be possible to use this form of the first
main theorem in the further presentation, since it demands
knowledge of differential calculus, which is important as a basis
of the computations of many theoretical predictions of *Thermodynamics*. It yields different equations
involving observable quantities and thereby leads to experimental
testing of the first main theorem.

Starting
from *dQ = dU + pdV *and using the theorem that the inner
energy of an ideal gas only depends on its temperature, you can
obtain (by means of simple computations of calculus) important thermal
properties of gases: If *c*_{p}* *and
*c*_{v}* *are the specific heats
at constant pressure and constant temperature, you have for every
gas *c*_{p}* - c*_{v }=
*R*/*m*, where *R *is the gas constant and *m
*the molecular weight. Hence the difference of the specific
heats of an ideal gas is constant, the difference of the
molecular heats *C*_{v}* = mc*_{v}*
*and *C*_{p}* = mc*_{p }is
even the same for *all*
*gase*s: mc_{p}
- mc_{v }= R.

Moreover the computation shows
that also *c*_{p }and* c*_{v
}depend only on the temperature and not on the volume, a
fact confirmed by the first measurements of Regnault. However, according to measurements, c_{p }is
also independent of the *temperature* over a wide temperature range. But then
this must also be valid in the same range for c_{v }(since
*c*_{p}* - c*_{v }=
const). Planck therefore completed the definition of
the *ideal gas* by the demand that his *c*_{p}
and* *his* c*_{v }are *completely** *independent of the temperature
and volume.

The computation also yields an
increase in temperature of a gas during *adiabatic *compression and a decrease during
adiabatic expansion. (A change of volume without heat input or
output is called *adiabatic *(Greek: a, dia, bainw = go; heat does
not pass through), that is, the *heat** *content
remains constant; an *isothermal** *change of volume takes place
at constant temperature, that is, with input and output of heat.)
Moreover (as accompaniment of *pv* = const during *isothermal** *change of volume) *pv*^{g} = const, where *g = **c*_{p}*/c*_{v
}during an *adiabatic** *change of volume. A comparison
of this equation with *pv* = const shows that the volume
of a gas during adiabatic compression decreases *more slowly** *(as a result of its increase in temperature)
than during isothermal compression. The adiabatic curves (Gibbs calls them *isentropic*)
in the *pv*-plane therefore drop more steeply to the *v*-axis
than the *isotherms *(hyperbolas) (Fig. 360).

Adiabatic compaction and
dilution of a gas are decisive during the propagation of sound in
a gas. In this context, we are only interested in air. Compaction
and dilution occur every second in a sound wave hundred, even
thousand times. The accompanying equally fast warming and cooling
of the air cannot adjust due to the rate of these changes, quite
apart from the fact that air conducts heat very badly. Hence
compaction and dilution in sound waves take place *adiabatically*. The formula *v = *(*p*/*d*)^{1/2}
for the velocity of propagation of longitudinal waves in a gas
contains the elasticity *p* of the gas. But a gas has two
kinds of elasticity, depending on whether its compression is
isothermal or adiabatic. The elasticity, measured adiabatically,
is larger than the isothermally measured one, in fact, in the
ratio of the specific heats *c*_{p}*/c*_{v
}= *k*, whence in the formula for the velocity
of sound, which involves the adiabatically measured ratio, *v
= *(*k**p*/*d*)^{1/2}
(Laplace 1816). With this correction of the formula *v = *(*p*/*d*)^{1/2},
due to Newton , the computed velocity of sound agrees
with the measured one.