heat and thermodynamics - iii / dr. mathivanan velumani
TRANSCRIPT
Ideal gas obeys gas Law ie, PV = RT.
Real gas doesn’t obey gas law
Ideal gas obey Boyle’s law P∝ 1/V at constant
temperature
Real gas doesn’t obey Boyle’s law, below critical
temperature and at high pressure
Boyle’s law doesn’t tell about the liquifaction of gas
below critical temperature.
Reganault in 1847, carried out experiment with real
gas at a pressure of 4 x 107 Pascals.
Amagat carried out experiment with gases at a
pressure of 3 x 108 Pascals in coal mines.
They found that at critical temperature and at high
pressure, gases can be liquified.
AN EXPERIMENT REGARDING LIQUEFACTION OF
GASES AND CRITICAL TEMPERATURE
A manometer is a device which uses
U-tube with one end fitted with a
glass bulb.
Mercury is filled in the manometer.
A liquid of low density is placed on
the top layer of the mercury in the
bulb.
The bulb is filled with vapor.
The whole arrangement is placed in
the constant temperature bath.
The temperature is varied using the
constant temperature bath.
At a particular temperature, there is
no difference seen between the liquid
and the vapor.
This particular temperature is called
critical temperature.
Thus gases can be liquefied below
this temperature at high pressure.
ANDREW’S EXPERIMENT
As the compression screws were applied, the pressure in the water increased and was transmitted equally to the two gas volumes.
By measuring the change in length of the air column, the air pressure (which was the same as the carbon dioxide pressure) was found.
The capillary tubes were very strong and Andrews obtained results up to pressures of 107 Pa.
Above about 50 oC Boyle's law was fairly closely obeyed.
As you can see the behavior of the 'gas' is different above and below about 30 oC - in fact Andrews found that the critical temperature for carbon dioxide was 30.9 oC.
below this temperature an increase in pressure would finally result in liquid carbon dioxide.
At the critical temperature, the gas and liquid are in equilibrium.
LIQUEFACTION OF GASES
As the gas is cooled at a constant pressure the molecules slow
down, and the gas occupies a smaller volume. As the average
distance between molecules becomes smaller, the forces of
attraction between molecules become great enough for them to be
bound together in a loose sort of way. The gas has become a liquid.
At high temperatures the kinetic energies (the energies of
movement) of the molecules are big enough for the forces of
attraction to be overcome. When the kinetic energies of the
molecules are reduced with the drop in temperature, the molecules
can hold together in the liquid.
Apart from cooling a gas, another method of liquefying it is to apply a
high pressure.
Enthalpy: Enthalpy is the amount of heat content used or released in a system at constant
pressure (or) energy released or absorbed during a chemical reaction.
Enthalpy: is a defined thermodynamic potential, designated by the letter "H", that consists of
the internal energy of the system (U) plus the product of pressure (p) and volume (V)
of the system.
Entropy: Degree of disorderness of the system.
Internal energy E: which is the sum of the kinetic and potential energies of the particles that
form the system.
Helmholtz Free Energy
Four quantities called "thermodynamic potentials" are useful in the chemical thermodynamics of
reactions and non-cyclic processes. They are internal energy, the enthalpy, the Helmholtz free
energy and the Gibbs free energy. The Helmholtz free energy F is defined by
Gibbs Free Energy
Four quantities called "thermodynamic potentials" are useful in the chemical thermodynamics of
reactions and non-cyclic processes. They are internal energy, the enthalpy, the Helmholtz free
energy and the Gibbs free energy. The Gibbs free energy G is defined by
Relating Helmholtz Energy to Gibbs Energy The four thermodynamic potentials are related by offsets of the "energy from the environment"
term TS and the "expansion work" term PV. A mnemonic diagram suggested by Schroeder can
help you keep track of the relationships between the four thermodynamic potentials.
The Helmholtz Energy is given by the equation:
A=U−TS
It is comparable to Gibbs Energy in this way:
G=A+PV
Gibbs free energy
The Gibbs Energy is named after a Josiah William Gibbs, an American physicist in the late 19th
century who greatly advanced thermodynamics; his work now serves as a foundation for this
branch of science. This energy can be said to be the greatest amount of work (other than
expansion work) a system can do on its surroundings, when it operates at a constant pressure and
temperature.
First, a modeling of the Gibbs Energy by way of equation:
∆G=U+PV−TS
Where:
U= Internal Energy
TS = absolute temperature x final entropy
PV = pressure x volume
Of course, we know that U+PV can also be defined as:
U+PV=∆H
Where:
∆H is change in enthalpy
Which leads us to a form of how the Gibbs Energy is related to enthalpy:
ΔG= ∆H−TS
All of the members on the right side of this equation are state functions, so G is a state function
as well. The change in G is simply:
ΔG=ΔH−TΔS
∆𝐺 - Gibbs free energy (available energy)
∆𝐻 - Change in enthalpy (change in Potential energy – enthalpy is the change in energy
possessed by the system)
T – Temperature
ΔS – change in entropy
Case 1: Change in enthalpy for spontaneous process – A boy sliding down in a slide – Here
Potential energy decreases and hence the enthalpy decreases
H H
Case 2: Change in entropy for spontaneous process: Diffusion of gases – when gases enter in to
another medium, the randomness increases and thereby the entropy increases
s S
Case 3: Change in temperature for spontaneous process : Inflated balloon at room
temperature doesn’t burst – when it is heated, the temperature of the gas molecules
get increased and due to expansion of gases inside the balloon, it gets burst.
T T
Substituting the values of above three cases in the first eqn, the Gibbs energy will be negative
Ie, ∆𝐺 < 0
In the case of oxidation of glucose in our body, brings carbon dioxide and water.
Here, the Gibbs energy will be positive
Ie, ∆𝐺 > 0
Reasoning Behind the Equation
As a quick note, let it be said that the name "free energy", other than being confused with another
energy exactly termed, is also somewhat of a misnomer. The multiple meanings of the word
"free" can make it seem as if energy can be transferred at no cost; in fact, the word "free" was
used to refer to what cost the system was free to pay, in the form of turning energy into work.
ΔG
is useful because it can tell us how a system, when we're given only information on it, will act.
ΔG<0
indicates a spontaneous* change to occur.
ΔG>0
indicates an absence of spontaneousness.
ΔG=0
indicates a system at equilibrium.
It was briefly mentioned that ΔG is the energy available to be converted to work. The definition
is self evident from the equation.
1) Calculate Hrxn for the following reactions given the following Hf values:
Hf (SO2, g) = 297 kJ/mol ; Hf (SO3, g) = 396 kJ/mol; Hf (H2SO4, l) = 814 kJ/mol
Hf (H2SO4, aq) = 908 kJ/molHf (H2O, l) = 286 kJ/mol Hf (H2S, g) = 20 kJ/mol
a) S(s) + O2(g) SO2(g) b) 2SO2(g) + O2(g) 2SO3(g)
c) SO3(g) + H2O(l) H2SO4(l) d) 2H2S(g) + 3O2(g) 2SO2(g) + 2H2O(l)
a) Change in enthalpy = Final enthalpy – Initial enthalpy
= Hf (SO2, g) - H𝑓 S, s + H𝑓 (O2, 𝑔)
= -297 – 0 - 0
= -297 KJ/mol.
The reaction is exothermic.
b) Change in enthalpy = Final enthalpy – Initial enthalpy
= Hf (2SO3, g) - H𝑓 2SO2, s + H𝑓 (O2, 𝑔)
= -396 – (2 x -297) - 0
= (2 x-396 +594) KJ/mol.
= -792+594
= -198 KJ/mol.
The reaction is exothermic.
c) Change in enthalpy = Final enthalpy – Initial enthalpy
= Hf (H2SO4, g) - H𝑓 SO3, s + H𝑓 (H2O, 𝑔) = -814 – (-396 -286)
= (-814 +682) KJ/mol.
= - 132 KJ/mol.
The reaction is exothermic.
d) Change in enthalpy = Final enthalpy – Initial enthalpy
= Hf (2SO2,g + 2H2O, l) - H𝑓 2H2S, g + H𝑓 (3O2, 𝑔)
= (2 x -297) + (2 x -286) - (2 x -20) + 0
= (-594 -572) KJ/mol.
= -1166 KJ/mol.
The reaction is exothermic.
1) Use Amagat’s law for ideal gas mixtures to calculate
VN2; VO2; Vtotal and
PN2;PO2; Ptotal.
Given: 21 mol% O2 and 79 mol % N2 at 25o C and 1 atm where R = 0.08205746.
Ie, n O2 = 0.21 ; n N2 = 0.79 ; T = 25 + 273 = 298K.
To find VN2 :
PV = nRT
Therefore, PVN2 = nN2 RT
VN2 = nN2 RT/P
VN2 = 0.79 x 0.08205746 x 298/1
VN2 = 19.31 Liters
To find VO2 :
PV = nRT
Therefore, PVO2 = nO2 RT
VO2 = nO2 RT/P
VO2 = 0.21 x 0.08205746 x 298/1
VN2 = 5.13 Liters
To find VTotal
VTotal = VN2 + VO2
= 19.31+5.13
= 24.44 Liters.
To find PN2 :
P N2V = nN2 RT
PN2 = nN2 RT/V
PN2 = 0.79 x 0.08205746 x 298/24.44
PN2 = 0.79 atm
To find PO2 :
P O2V = nO2 RT
PO2 = nO2 RT/V
PO2 = 0.21 x 0.08205746 x 298/24.44
PO2 = 0.21 atm
To find PTotal
PTotal = PN2 + PO2
= 0.79 + 0.21
= 1 atm.
Vander waal’s equation of state
The ideal gas equation is PV = RT
Here, the size of the gas molecules were
considered to be negligible and the inter-
molecular force of attraction are absent.
But, in practice, at high pressure, the size of
gas molecules becomes significant when
compared to its volume.
Also, at high pressure, the molecules come
closer to each other and inter-molecular force of
attraction becomes appreciable.
Hence Vander-waals made some correction for
pressure and volume in the ideal gas equation.
Correction for Pressure:
The correction for pressure depends upon
•The number of molecules striking the walls of the
container/sec.
•The number molecules present/unit volume.
Both these factors depends on density
ρ = m/v
Therefore, P ∝ 1
𝑉2
Or
P = 𝑎
𝑉2 _______________ (1)
Correction for volume
Volume of each molecule = 4
3 𝜋𝑟3= x ___________ (2)
The sphere of influence of the molecule with respect to the
nearest molecules
S = 4
3 𝜋 (2r)
3
S = 8 ( 4
3 𝜋𝑟3)
using eqn 2,
S = 8x _____________ (3)
Let the volume occupied by the first molecule = V
The volume occupied by second molecule = V- S
The volume occupied by third molecule = V – 2S
………………………………………………….
The volume occupied by the nth molecule = V- (n-1) S
Therefore, the average volume occupied by the molecules
= 𝑛𝑉
𝑛 -
𝑆
𝑛 1 + 2 + …………… + (n − 1)
= V- 𝑆
𝑛
𝑛(n−1)
2
= V- 𝑆𝑛
2 -
𝑆
2
Let S/2 = 0
= V- 𝑆𝑛
2
We know S = 8x from equation 3.
Substituting the value of S in the above equation,
= V- 8𝑥𝑛
2
= V- 4xn
= V-b __________________ (4)
Where b = 4xn
Therefore, equation 4 is the correction for volume.
Therefore, Vanderwaals gas equation becomes,
( P + 𝑎
𝑉2 ) (V-b) = RT
Reduced equation of state
Cubic equations of state are called such
because they can be rewritten as a cubic
function of Vm.
The Van der Waals equation of state may
be written:
With the reduced state variables, i.e.
Vr=Vm/Vc ________________(4) (Vr -relative
volume)
Pr = P/Pc ___________(5) (Pr -relative
pressure)
Tr=T/Tc ________________(6) (Tr -relative
temperature)
THE REDUCED FORM OF THE VAN DER WAALS EQUATION CAN BE
FORMULATED:
SUBSTITUTING EQUATIONS 1, 2,3,4,5 AND 6 IN A,
The above equation is the reduced Vander waal’s equation of
state
Critical Constants
The critical temperature, Tc, is characteristic of every gas and may be
defined as: “The temperature below which the continuous increase of
pressure on a gas ultimately brings about liquefaction and above which
no liquefaction can take place no matter what so ever pressure be
applied”.
The pressure required to liquefy the gas at critical temperature is called
critical pressure and the volume occupied by 1 mole of gas under these
conditions is called the critical volume.
Condition to find critical constants
𝜕𝑃
𝜕𝑉 T = 0 and
𝝏𝟐𝑷
𝝏𝑽𝟐 T = 0
If the solution of the above equations becomes zero or infinity then there
exists a critical point. Critical point is the point at which the gas changes
its state with respect to change in pressure and volume.
Critical Temperature Tc = 𝟖𝒂
𝟐𝟕 𝑹𝒃
Critical pressure Pc = 𝒂
𝟐𝟕 𝒃𝟐
Critical Volume Vc = 3b
Temperature
Temperature is a measure the sensation of warmth or coldness of an object, felt from contact
with it. This sensation of touch gives an approximate or relative measure of the temperature.
Temperature is measured in different scales, including Fahrenheit (F) and Celsius (or centigrade,
C). The units of the Fahrenheit and Celsius scales are called degrees and are denoted by °.
Swedish astronomer Anders Celsius devised the Celsius scale in 1742. He fixed the 0° of the
scale at the freezing of water, and the 100° at the boiling of water.
Themometer
A thermometer is used to measure the temperature of an object – it is used to find how cold or
hot the object is. Galileo invented a rudimentary water thermometer in 1593. He called this
device a "thermoscope". However, this form was ineffective as water freezes at low
temperatures.
In 1714, Gabriel Fahrenheit invented the mercury thermometer, the modern thermometer. The
long narrow uniform glass tube is called the stem of a thermometer. The small tube called the
bulb, which contains mercury. Mercury is toxic, and it is very difficult to dispose it when the
thermometer breaks. So, nowadays digital thermometers are used to measure the temperature, as
they do not contain mercury.
Types of Thermometers
There are different types of thermometers that measure the temperatures of different things like
air, our bodies, food and many other things. There are clinical thermometers, laboratory
thermometers, Galileo thermometers and digital remote thermometers. Among these, the
commonly used thermometers are clinical thermometers and laboratory thermometers.
Clinical Thermometer:
These thermometers are used to measure the temperature of the human body, at home, clinics
and hospitals. All clinical thermometers have a kink that prevents the mercury from falling down
rapidly so that the temperature can be noted conveniently. There are temperature scales on either
side of the mercury thread, one in Celsius scale and the other in Fahrenheit scale.
A clinical thermometer indicates temperatures from 35° C to 42° C or from 94° F to 108° F.
To note a reading, place the thermometer in the person’s mouth. Since the Fahrenheit scale is
more sensitive than the Celsius scale, body temperature is measured in degrees Fahrenheit only.
A healthy person’s average body temperature is between 98.6° F and 98.8° F .
Precautions:
• Wash the thermometer before and after use with an antiseptic solution, and handle it with
care.
• See that the mercury levels are below the kink and don’t hold the thermometer near its bulb.
• While noting down the reading in the thermometer, place the mercury level along the eye
sight.
• Do not place the thermometer in a hot flame or in the hot sun.
Laboratory Thermometers
These thermometers are used to measure the temperature in school and other laboratories for
scientific research. They are also used in the industry as they can measure temperatures higher
than what clinical thermometers can record. The stem and the bulb are longer when compared to
that of a clinical thermometer. A laboratory thermometer has only the Celsius scale ranging from
-10o C to 110
o C.
Precautions:
• A laboratory thermometer doesn’t have a kink.
• Do not tilt the thermometer. Place it upright.
• Note the reading only when the bulb has been surrounded by the substance from all sides.
Heat Temparature
1. Heat is a form of energy obtained
due to random motion of molecules in
a substance.
2. The S.I. unit of heat is joule (J).
3. The amount of heat contained in a
body depends on temperature, mass
and material of body.
4. Heat is measured using the principle
of calorimetry.
5. Two bodies having same quantity of
heat may deffer in thier temperature.
6. When two bodies areplaced in
contact, the total amount of heat is
equal to the sum of heat of individual
body.
1. Temperature is a quantity which
determaines the direction of flow of heat
on keeping the two bodies at different
temperatures in contact.
2. The S.I. unit of temperature is kelvin
(K).
3. The temperature of a body depends on
the average kinetic energy due to random
motion of its molecules.
4. Temperature is a measured by a
thermometer directly.
5. Two bodies at same temperature may
differ in the quantities of heat contained
in them.
6. When two bodies at different
temperatures are placed in contact, the
resultant temperature is a temperature in
between the two temperatures.
Mercury as a Thermometric Liquid:
Mercury fulfils practically all the requisites of a thermometric liquid as the following:
Mercury does not stick to the sides of the glass.
Mercury exerts very low vapour pressure.
Mercury is a good conductor of heat.
Mercury has low specific heat capacity.
Mercury expands uniformly.
Mercury is easily available in pure state..
Mercury is an opaque and shinning liquid metal.
IMercury has a high b.p. (357°C) and low m.p (- 39°C)
Disadvantages of mercury as Thermometric Liquid:
Mercury freezes below -39°C and hence, it cannot be used in very cold regions like
Antarctic or Arctic.
Mercury's expansion is not very large for 1°C rise in temperature and hence, very small
changes in temperature cannot be measured.
Alcohol as Thermometric Liquid:
Alcohol can be coloured brightly and hence, is easily visible.
Alcohol freezing point is below -100°C and hence, can record very low temperatures.
Alcohol's expansion per degree centigrade rise in temperature is very large and hence,
very sensitive thermometers can be made with it.
Disadvantages of Alcohol as Thermometric Liquid:
Alcohol has a high vapour pressure.
Alcohol sticks to the sides of glass.
Alcohol has high specific heat capacity.
Alcohol can not be used for measuring high level temperatures as alcohol boils at 78°C.
Alcohol is difficult to obtain pure alcohol.
Alcohol is not good conductor of heat.
Disadvantages of Water as Thermometric Liquid:
Water sticks to the sides of glass.
Water is transperant.
Water evaporates under vaccum conditions.
Water does not expand uniformly.
Water has highest specific heat capacity (4.2 J/gK)
Expansion of water per degree rise in temperature is very small.
Water cannot be obtained in pure form easily.
Melting point of water is 0°C and boiling point is 100°C . Thus , the temperatures less
than 0°C and more than 100°C cannot be measured.
Water is a bad conductor of heat.
The Constant-Volume Gas Thermometer
and the Absolute Temperature Scale
A typical graph of pressure versus temperature taken
with a constant-volume gas thermometer. The two dots
represent known reference temperatures (the ice and
steam points of water
Thus, the conversion between these temperatures is
where TC is the Celsius temperature and T is the absolute temperature.
The value of P is used to calculate the temperature. Temperature is calculated using the
formula
T = aP + b Where a is the constant of ice at 0oC and b is constant of steam at 100oC.
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