Physics Program- Lecture notes

1. Ideal gases are gases of low pressure, or gases that have a low density.
• We use ideal gases to approximate the more complex relationships that can
exist.
• Relationships between the pressure, volume and temperature are called
equations of state.
• These equations can be very complicated but relationship between pressure ,
volume and temperature can be found for ideal gases and is much simpler
than the more complex general equations of state.
Kinetic Theory of GasesKinetic Theory of Gases
• A theory that attempts to explain the behavior of an ideal gas.
•An ideal gas can be defined as: The gas that obeys the assumptions of
the kinetic theory of gases.

2. The assumptions Kinetic Theory
• Gas consists of large number of tiny particles (atoms or molecules)
with freedom of movement
• There exist no external forces (density constant) and no forces
between particles except when they collide.
• Particles, on average, separated by distances large compared to
their diameters
• Particles make elastic collisions with each other and with walls of
container
• The average kinetic energy of the gas particles is directly
proportional to the Kelvin temperature of the gas

3. KINETIC THEORY
( ) ( )
L
mv
vL
mvmv 2
2
collisionssuccessivebetweenTime
momentumInitial-momentumFinal
forceAverage
−
=
+−−
=
=
From Newton’s Law
The internal energy of monatomic ideal gas

4. L
mv
F
2
=
For a single molecule, the average force is:
For N molecules, the average force is:














=
L
vmN
F
2
3 root-mean-square
speed
V: is the volume
Hence, The pressure is given by:

5. ( ) ( )2
2
1
3
22
3
1
rmsrms mvNmvNPV ==
2 31
2 2KE rms Bmv k T= =
We can relate the state equation of the ideal gas with its
average kinetic energy as
The internal energy of monatomic ideal gas
3 3
2 2BU N k T nRT= =
PV BNk T⇒ =
kB = R/NA
& / An N N= PV nRT⇒ =But

6. The Equation of State of Ideal Gases
P – pressure [N/m2
]
V – volume [m3
]
n – number of moles of gas
T – the temperature in Kelvins [K]
R – a universal constant
nRTPV =
Kmol
J
R
⋅
≈ 31.8
The ideal gas
equation of state:
An equation that relates macroscopic variables (e.g., P, V, and T) for a given
substance in thermodynamic equilibrium.
In equilibrium (≡ no macroscopic motion), just a few macroscopic parameters are
required to describe the state of a system.
f (P,V,T) = 0
Geometrical
representation of the
equation of state:
P
V
T
an equilibrium
state
the equation-
of-state surface
R= 0.08214 L atm/mol K

7. Mm – Molar mass [g/mol]
N – Number of molecules [molecules]
NA – Avogadro’s Number of molecules per mole
[molecules/mol]
mM
m
n =
AN
N
n =
Where,TNkPV B=
the Boltzmann constant
kB = R/NA ≈ 1.38⋅10-23
J/K
(introduced by Planck in 1899)
Avogadro’s Law: equal volumes of different gases at the same P
and T contain the same amount of molecules.
Ideal gas, constant mass (fixed quantity of gas)
1 1 2 2
1 2
PV P V
T T
=

8. Boyle's Law (constant temperature)
P = constant / V
Charles Law (constant pressure)
V = constant × T
Gay-Lussac’s Law (constant volume)
P = constant × T
Important laws

9. Isothermals pV = constant
0
20
40
60
80
100
120
140
160
180
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
volume V (m
3
)
pressurep(kPa)
100 K
200 K
300 K
400 K
n RT
p
V
=

10. Energy in terms of Temperature
At T = 300K
3
2
BKE k T=
The kinetic energy is proportional to the temperature,
and the Boltzmann constant kB is the coefficient of
proportionality that provides one-to-one correspondence
between the units of energy and temperature.
If the temperature is measured in Kelvins, and the energy – in Joules:
kB = 1.38×10-23
J/K
23
19
1.38 10 300
26
1.6 10
Bk T meV
−
−
× ×
= ≈
×

11. In General
The internal energy of an ideal gas
with f degrees of freedom:
TNk
f
U B
2
=
f ⇒ 3 (monatomic), 5 (diatomic), 6 (polyatomic)
How does the internal energy of air in this (not-air-tight) room change
with T if the external P = const?
2 2
B
f f
U Nk T PV= =
does not change at all, an increase of the kinetic energy of individual
molecules with T is compensated by a decrease of their number.

12. Real Gases
General Observations
• Deviations from ideal gas law are particularly important
at high pressures and low temperatures
• Real gases differ from ideal gases in that there can be
interactions between molecules in the gas state
– Repulsive forces important only when molecules are nearly in
contact, i.e. very high pressures
• Gases at high pressures , gases less compressible
– Attractive forces operate at relatively long range (several
molecular diameters)
• Gases at moderate pressures are more compressible since
attractive forces dominate
– At low pressures, neither repulsive or attractive forces dominate
→ ideal behavior

13. Real GasesReal Gases
Deviations from IdealityDeviations from Ideality
J. van der Waals, 1837-1923,J. van der Waals, 1837-1923,
Professor of Physics, Amsterdam.Professor of Physics, Amsterdam.
Nobel Prize 1910.Nobel Prize 1910.
. The Ideal Gas Law ignores both the volume
occupied by the molecules of a gas and all
interactions between molecules, whether
attractive or repulsive
 In reality, all gases have a volume and the
molecules of real gases interact with one
another.
 For an ideal gas, a plot of PV/nRT versus P
gives a horizontal line with an intercept of 1 on
the PV/nRT axis.

14.  The reasons for the deviations from
ideality are:
• The molecules are very close to
one another, thus their volume is
important.
• The molecular interactions also
become important.
 Real gases behave ideally at ordinary temperatures and
pressures. At low temperatures and high pressures real gases
do not behave ideally.

15. Real Gases
Van Der Waals Equation
Real gases do not follow PV = nRT perfectly. The van der
Waals equation corrects for the nonideal nature of real
gases.
a corrects for interaction between atoms.
b corrects for volume occupied by atoms.
2
2
n a
V
(P )(V-nb) nRT+ =

16. Real Gases
Van Der Waals Equation
• A non-zero volume of molecules = “nb” (b is a constant
depending on the type of gas, the 'excluded volume‘, it
represents the volume occupied by “n” moles of molecules).
• The molecules have less free space to move around in, so
replace V in the ideal gas equation by V - nb
• Very roughly, b ∼ 4/3 πr3
where r is the molecular radius.

17. Real Gases :Van Der Waals Equation
The attractive forces between real molecules, which reduce the pressure:
 p ∝ wall collision frequency and
 p ∝ change in momentum at each collision.
Both factors are proportional to concentration, n/V, and p is reduced by
an amount a(n/V)2
, where a depends on the type of gas.
[Note: a/V2
is called the internal pressure of the gas].
2
2
n a
V
P becomes (P )+
n2
a/V2
represents the effect on pressure to
intermolecular attractions or repulsions.

18. If sulfur dioxide were an “ideal” gas, the pressure at 0°C exerted by 1.000
mol occupying 22.41 L would be 1.000 atm. Use the van der Waals
equation to estimate the “real” pressure.
Example:
Solving the van der Waals
equation for pressure.
2
2
nRT n a
P -
V-nb V
=
R= 0.0821 L.
atm/mol.
K
T = 273.2 K
V = 22.41 L
a = 6.865 L2.
atm/mol2
b = 0.05679 L/mol
L atm
mol K
(1.000 mol)(0.08206 )(273.2 )
P
22.41 L - (1.000 mol)(0.05679 L/mol)
K×
×
=
2
2
2 L atm
mol
2
(1.000 mol) (6.865 )
-
(22.41 L)
×
P 0.989 atm= The “real” pressure exerted by 1.00 mol of SO2 at
STP is slightly less than the “ideal” pressure.

19. Work and Heating (“Heat”)
We are often interested in ∆U , not U. ∆U is due to:
Q - energy flow between a system and its environment due to ∆T across a
boundary and a finite thermal conductivity of the boundary
– heating (Q > 0) ---------cooling (Q < 0)
W - any other kind of energy transfer across boundary ( work )
Heating/cooling processes:
conduction: the energy transfer by molecular contact – fast-
moving molecules transfer energy to slow-moving molecules by
collisions;
convection: by macroscopic motion of gas or liquid
radiation: by emission/absorption of electromagnetic radiation.
HEATING
WORK
Work and Heating are both defined to describe energy
transfer across a system boundary.