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Theory of thermodynamics

1.1 The goals of thermodynamics

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Thermodynamics is the branch of
physics that studies the macroscopic systems in which the thermal affects are
taken into account.  These systems are
considered to be in equilibrium.  When
the systems are in equilibrium it is easier to study them both experimentally
and theorically.

A system is a thermodynamic
event such as some liquid that is inside a closed container for which we have
to measure the pressure volume temperature and other physical properties.  The system has also its surroundings, which
means that it is not enough to know the system in order to get accurate values
for the quantities involved in the study. 
We have also to take into consideration even the surroundings in order
to the study the situation described.

When studying thermodynamics we
often neglect the interaction between particles.  For this reason we are more interested in
macroscopic systems rather than in microscopic ones.  So, thermodynamics represents a very general
approach for the study of a System Properties. 
However, with a good approximation it is very useful when studying the
daily life phenomenon regarding properties such as heat, temperature, thermal
expansion and contraction, pressure, volume, internal energy etc.

1.2 The universe and its components

First let’s give some
definitions:

The thermodynamic system is confined
always with some boundary.  The external
part of this boundary is called surroundings as we said before.  So, everything that exists is either a part
of the system, a boundary or surroundings. 
Hence, we can say that the system plus surroundings plus boundaries
gives the universe.

                                  

We must not neglect the
importance of boundary.  It is equally
important as the system and the surroundings. 
Without the boundary we can’t have heat-exchange, thermal interactions
etc.  The boundary is called adiabatic when
it prevents any exchange of heat between the system and the surroundings.  On the other hand, the diathermal boundary
the walls heat exchange between the system and its surroundings.

In Thermodynamics there are
three types of systems: open systems, closed systems and insulated
systems.  In open systems will particles
are thermally interacting.

1.3 Equilibrium

A system is in equilibrium when
its physical properties do not change with time.  For example, when no heat is supplied to some
liquid, its temperature will remain constant all the time.  The liquid will have the same temperature as
the environment if no heat is supplied to it. 
On the other hand, when heat is given to the liquid, its mass will
decrease because some liquid will turn into a vapor.  So this system is not anymore in equilibrium.

How can this equilibrium be
reached?  Let’s consider some gas which
initially is allowed to slowly fill a box. 
As long as the gas is filling the box, the system is not in equilibrium.  This equilibrium will be reached after the
gas flow is stopped and the gas fills all the volume of the box.  Now the system is in equilibrium.  The gas will continue to flow within the box
but this is a microscopic process and we are not interested in that.  And as we said before, we are interested in macroscopic
processes. 

Imagine if we are measuring the
pressure in two different moments.  First
we measure the pressure during the gas entrance into the box.  In this moment the pressure is changeable
because the gas flow is not uniform.  If
we measure of the gas pressure after the gas has stopped flowing inside the
box, we will find that the pressure is almost constant.  So, we can conclude that the pressure is
uniform only when the equilibrium of the system has been reached.

                                                     

Therefore, equilibrium is
verified in two conditions: the absence of macroscopic flows and the constant
pressure.  If the gas pressure is not
uniform within the box, so that the pressure in one side of the box is not
equal to the pressure on the other side of net force will appear as P = F / A

where A it is the side area of the box.  Therefore

            F = (P2 – P1) x
A

Hence, if there is a net force
there is no equilibrium and this comes because of non-uniform pressure.                 

   

 So, we obtain an important property of the system:

“Non -uniform pressure results
in macroscopic flow and this one results in absence of equilibrium.”

And also we can say that equilibrium
is reached only at constant pressure.

1.4 Thermodynamic variables

A variable is a quantity that
can change during the process in thermodynamics.  There are several variables in thermodynamics
which characterize the system as a whole. 
The values that these variables will take during the process will not
depend on the place where these values are measured.  The thermodynamic variables can be defined
only when the system is in equilibrium. 
Variables are necessary to describe the physical state of the thermodynamic
system in equilibrium.

Let’s take an example with a
fluid.  There are two variables that have
to be considered when studying this system. 
These two variables are pressure and volume.  There are other thermodynamic variables such
as temperature, entropy, and internal energy etc..  They are common to all thermodynamic systems.

Thermodynamic variables can be
divided into two main categories: extensive and intensive variables.  Let’s imagine that a thermodynamic system is
divided into two equal parts where each of these parts carries its own
thermodynamic variables.  The question is:
how to the variables of the splitted system, compare to the variables of the
original system?  Some quantities women
be affected while some others will be halved. 
The quantities that are not affected are intensive variables while the
others are extensive variables.  For
example the pressure is not affected by the division, so it is an intensive up
variable, but the volume has been halved, so the volume is an extensive
variable.

1.5 Zeroth law of thermodynamics

Let’s imagine that the systems
A and B and are placed on both sides of the diathermal wall so that heat is
allowed to be exchanged.  These systems
are in thermal contact with each other.

Let’s consider both systems
initially in equilibrium but these equilibriums are internal equilibriums of
the separate systems and not equilibrium of the joint system.  When we placed in contact both objects, the
previous equilibriums will change until both objects will reach a new equilibrium.  In this case, the physical properties of
objects A and B had changed and the new equilibrium has established new values
of their physical properties.  This new
equilibrium of both systems is known as thermal equilibrium which appears when
two systems are brought in thermal contact. 
This fact is known as the first principle of thermodynamics or zeroth
law of thermodynamics.  It states that

When a
system A is in thermal equilibrium with a system B, and the system B is in
thermal equilibrium with a system C, the system A is also in equilibrium with
the system C.

1.6 Temperature

We can feel the difference
between a hot object and a cold object. We may say that the temperature is the degree
of hotness of an object. But this definition is intuitive and makes reference
to our senses. We cannot use this approach to assign a numerical value to temperature.
A cup of tea can be “warm” according to one person and “very
hot” according to another. We must find a universally applicable way to
measure the numerical value of temperature. The device used to measure the
temperature is called a “thermometer”. There are different types of
thermometers utilizing different physics laws as their working principle. It is
possible to measure the temperature by using resistance of a wire, expansion properties
of solids, or even an infrared camera. But the common thermometer we are so familiar
with utilizes expansion of liquids to measure the temperature.

1.7
Scales of temperature

The mercury thermometer was invented by Gabriel D. Fahrenheit. It
consists of a thin glass tube sealed at both ends (capillary tube) and partly
filled with mercury. Above the mercury column is vacuum, for the mercury to
expand freely. As the temperature increases the mercury column in the tubes
rises. Assigning numerical values to different heights of the mercury column is
called “calibrating the thermometer”. Using different criteria for
calibration gives rise to different temperature scales. Two commonly used
temperature scales in daily life are Celsius and Fahrenheit scales. Swedish
astronomer Celsius used the freezing and boiling points of water under 1 atm pressure
as the reference points for his scale.

Temperature of ice-water mixture is taken as 0 °C and temperature
of boiling water is taken as 100 °C in Celsius scale. Between these two
reference points is divided into 100 equal parts. This is why the Celsius scale
is also called centigrade (=100 grades) scale. German
physicist Fahrenheit
used the temperatures of hottest and
coldest days in his country to calibrate his scale.

In scientific studies Kelvin scale is used to express temperature.
Kelvin scale is also named as “absolute temperature scale”. Later in
this chapter we will learn further about Kelvin scale.

The correspondence
among the three temperature scales is given below.

Between
freezing and boiling points of water, temperature increases from 0 to 100 in
Celsius scale, from 273.15 to 373.15 in Kelvin scale and from 32 to 212 in Fahrenheit
scale. Consequently the conversion formulas must be

 

            TK
= TC + 273.15   
and  

            Capillary tube thermometers have
a problem. Two thermometers calibrated in the same way gives us different temperature
readings, if the liquids used in the thermometers are different. Suppose
freezing and boiling points of water are marked on a mercury thermometer and a
glycerin thermometer. The two thermometers agree on 0 °C and 100 °C, but they
will show different values for intermediate temperatures. This effect is caused
by the different expansion properties of different liquids.

To
overcome this difficulty and reach a universal agreement on the full range of temperature
scale, “constant volume gas thermometer” is used. This thermometer is
also used in defining the Kelvin scale.

1.8
Thermal Expansion

Solid, liquid, gas, all substances expand (become larger) with
increasing temperature. Power lines sagging in the summer and tightening in the
winter is a common observation. We will investigate three types of expansion: linear,
area and volume expansion.

For long and thin objects like long copper wire, change in
thickness is very small

and can be neglected.
If we assume the object to be one dimensional, expansion

is linear.

a)
Linear expansion

Change in length depends on three factors:

– Change in temperature (DT)

– Initial length (L0)

– Material of the object (?)

? is called the “linear expansion coefficient” and is
different for each material. Consider a thin long steel rod.

 

 

                  

        The
linear expansion coefficient is defined as the fractional change in length (L) per
unit temperature change (DT). Thus,    

Hence, the change in length is given by

 

And the new length of the rod will be

 

            Or

b)
Area Expansion

Consider a thin, flat piece of metal. Suppose the area of the
object increases from A0 to Af   as
temperature increases from T0 to Tf. As usual,
temperature change is DT=Tf  – T0 and area change is DA= Af  – A0.

The dependence of area change on temperature change is given by:

 

Because for each dimension the linear expansion coefficient is , and as we have 2 dimensions involved here, we must
write 2? instead of .

The  in
the formula is the same as in the linear expansion formula.

                        

c)
Volume Expansion

Following the same line of thought, we can draw that for a 3
dimensional object, expansion, the formula becomes:

                               

 

            1.9
Molecular Interpretation of Temperature

We had
defined temperature as degree of hotness. What is the physical difference between
a hot object and a cold object aside from feeling hot or cold? In other words,
which property of a substance determines its temperature?

Molecular – kinetic theory tells
us that temperature of a substance is related to the motion of molecules
forming the substance. Remember from previous chapter that all matter (solid,
liquid, gas) consist of unthinkably small molecules, and these molecules are in
continuous random motion which is called thermal motion. The form of thermal
motion depends on the phase of the substance. In a solid, molecules oscillate
around fixed positions. In a gas, they fly around freely continuously colliding
with each other.

Temperature
of a substance is related to thermal motion of molecules. In general terms, higher
temperature means faster moving molecules.

For
solids and liquids the relation between molecular speeds and temperature of substance
is complicated because of the bonds between the molecules. In case of an ideal
gas this relation is quite simple. For an ideal gas, temperature is
a measure of average kinetic energy of molecules of the gas. Average
kinetic energy means sum of kinetic energies of molecules divided by number of
molecules. Actually average kinetic energy of gas molecules is a very small
number since the mass of a molecule is so small. At 20 °C for example an oxygen
molecule in the air has about 10-20 J of kinetic energy.

As we know,
molecules of a gas can have a wide range of velocities, and speed of a molecule
changes millions of times in a second due to the collisions with other molecules.
At a given instant one molecule of a gas sample may nearly be at rest while
another molecule moves with almost speed of light. Consequently kinetic energies
of molecules of a gas are quite different. But average kinetic energy of molecules
gives us an idea about the behavior of the gas sample. An analogy to average
kinetic energy of gas molecules could be the national income per capita of a
nation. As we know, national income per capita means total value of all income
of a country in one year, divided by the population of the country. Consider
two countries X and Y. National income per capita of country X is higher than
that of Y. This means average citizen of X country is richer than average citizen
of Y country. But naturally in both countries we can find very rich and very
poor people. Higher income per person does not mean that every single person in
X country has much money, but still it gives us an idea about the average richness
level in the country.

Similarly
a gas sample with higher temperature has molecules with higher kinetic energy
on average than a colder sample. But it is always possible to find very fast and
very slow molecules in both samples. Later in this book we will be able to show
that

Average KE of gas molecules = (a
constant) x (temperature)

 

 

2. Gas laws

Gas
laws are relations between the macroscopic parameters of a gas. Macroscopic parameters
of a gas are pressure, volume, and temperature. These are called macroscopic
because we can either see them or directly measure them with devices such as
thermometer or manometer. On the other hand, a quantity like speed of a single
molecule is microscopic by definition.

Before
starting to deal with gas laws, we should get to know something called “cylinder
– piston system” better. We will be using this device throughout all our thermodynamics
studies.

Consider
a cylinder – piston system as described below:

The
piston is tight fitting enough not to allow any gas to escape, but in the same time
moves up and down without friction. Naturally this is a difficult apparatus to
produce. But it will be useful to get a grip on the basics of gas

We can adjust the pressure
applied on the inside by the piston, by changing the weight of the piston as
shown below. 

 

                                                  

 

In a
cylinder – piston system:

Movable
piston implies constant pressure.

Historically
the relations among the temperature pressure and volume of a gas were
established in 17th century by different scientists. Now we remember the
relations by the names of these scientists. We will investigate three gas laws
here, at constant temperature, at constant pressure, and at constant volume.

Pressure
volume relation of a gas sample at constant temperature is established nearly
at the same time by the English physicist Boyle and French scientist Mariotte. Now
we name the relation as Boyle-Mariotte law.

a.
Boyle-Mariotte Law

Pressure – volume relation of a gas sample at constant
temperature.

Changing volume and pressure of a gas sample at constant
temperature is called isothermal process. “Isothermal” means happening
at “constant temperature”. When a fixed amount of gas is compressed,
decreasing its volume, we expect its pressure to increase. You can easily
observe this effect by trying to squeeze the air in a syringe with its nozzle
closed. Pushing the piston will require more and more force as the volume of
the air trapped inside decreases. But this “syringe experiment” is
not a good example for isothermal process. As the air in the syringe is compressed,
not only the pressure but also the temperature increases. To overcome this
difficulty and keep the temperature constant we must work carefully.

                   

To achieve isothermal process we can make such an arrangement. We
place the cylinder in figure in a lake to keep the temperature constant. Then
to increase the pressure slowly, we start to drop grains of sand on the piston
one by one. Impact of each grain pushes the piston down a little and
temperature tries to increase by – say – 0.01 °C, but since we cannot heat the
whole lake, and the process is slow enough, the gas inside the cylinder will
have time to cool down, until its temperature equals the temperature of lake water.
We can say the temperature of the gas is kept constant.

The lake in the process described above is acting as a heat
reservoir. This means it is cooling down (or heating up) another object without
changing its own temperature.

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