PROPERTIES OF GASES (REPORT) Physical Chemistry 1 Undergraduate Lesson.ppt

Properties of Gases

1. Physical Chemistry I
Text: Physical Chemistry, 7th Edition, Peter Atkins and J. de Paula

2. Chapter 1: Properties of Gases

3. The State of Gases
Gases are the simplest state of matter
»Completely fills any container it occupies
»Pure gases (single component) or mixtures of components
Equation of state - equation that relates the variables defining its physical properties
»Equation of state for gas: p = f (T,V,n)
»Gases (pure) Properties - four, however, three specifies system
Pressure, p, force per unit area, N/m
2
= Pa (pascal)
Standard pressure = p
ø
= 10
5
Pa = 1bar

Measured by manometer (open or closed tube), p = p
external
+ gh
g = gravitational acceleration = 9.81 m/s
-2
Mechanical equilibrium - pressure on either side of movable wall will equalize
Volume, V
Amount of substance (number of moles), n
Temperature, T, indicates direction of flow of energy (heat) between two bodies;
change results in change of physical state of object
Boundaries between objects
»Diathermic - heat flows between bodies. Change of state occurs when bodies of
different temp. brought into contact
»Adiabatic - heat flows between bodies. No change of state occurs when bodies of
different temp brought into contact

4. Heat Flow and Thermal Equilibrium
Thermal equilibrium - no change of state occurs when two objects are in contact through a
diathermic boundary
Zeroth Law of Thermodynamics - If A is in thermal equilibrium with B and B is in
thermal equilibrium with C then A is in thermal equilibrium with C
»Justifies use of thermometer
»Temperature scales:
Celsius scale, , · (°C) degree defined by ice point and B.P. of water
Absolute scale, thermodynamic scale , (K not °K)
T (K) =  + 273.15
A B
Heat
Diathermic Wall
A B
Diathermic Wall
High
Temp.
Low
Temp.
A B
Adiabatic Wall
No Heat
No Heat
T
A
= T
B

5. Equation of State for Gases ( p = f(V,T,N)
Ideal (Perfect) Gas Law
Approximate equation of state for any gas
»Product of pressure and volume is proportional to product
of amount and temperature
PV = nRT
R, gas constant, 8.31447 JK
-1
mol
-1
R same for all gases, if not gas is not
behaving ideally

Increasingly exact as P  0 a limiting law
For fixed n and V, as T  0, P  0 linearly
»Special cases (historical precedent): Boyle’s Law (1661),
Charles' Law [Gay-Lussac’s Law (1802-08)]; Avogadro's
principle (1811)
Used to derive a range of relations in thermodynamics
»Practically important, e.g., at STP (T= 298.15, P = p
ø

=1bar), V/n (molar volume) = 24.789 L/mol
For a fixed amount of gas (n, constant) plot of properties of gas
give surface
»Isobar - pressure constant - line, V  T
»Isotherm - temperature constant, hyperbola, PV =
constant
»Isochor - volume constant - line P  T http://www.chem1.com/acad/webtext/gas/gas_2.html#PVT

6. Ideal (Perfect) Gas Law - Mixtures
Dalton’s Law: Pressure exerted by a mixture of gases is sum
of partial pressures of the gases
»Partial pressure is pressure component would exhibit if
it were in a container of the same volume alone
»p
total
= p
A
+ p
B
+ p
C
+ p
D
+ ……. (A, B, C, D are
individual gases in mixture)
 p
J
V = n
J
RT
This becomes :
If x
J
is the fraction of the molecule, J, in mixture {x
J
= n
J
/
n
Total
), then  x
J
=1
If x
J is the partial pressure of component J in the mixture, p
J
= x
J
p, where p is the total pressure
»Component J need not be ideal
»p =  p
J =  x
J p this is true of all gases, not just
ideal gases
p
total
n
J
J
RT
V
0 1
Mole Fraction B, x
B
P
p = p
A + p
B
p
B
= x
B
p
p
A
= x
A
p

7. Real Gases - General Observations
Deviations from ideal gas law are particularly important at
high pressures and low temperatures (rel. to condensation
point of gas)
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 (spn small), gases less compressible
»Attractive forces operate at relatively long range (several molecular
diameters)
Gases at moderate pressures (spn few molecular dia.) are more
compressible since attractive forces dominate
»At low pressures, neither repulsive or attractive forces dominate - ideal
behavior

8. Compression Factor, Z
Compression factor, Z, is ratio of the
actual molar volume of a gas to the molar
volume of an ideal gas at the same T & P
»Z = V
m/ V
m°, where V
m = V/n
Using ideal gas law, p V
m
= RTZ
The compression factor of a gas is a
measure of its deviation from ideality
»Depends on pressure (influence of
repulsive or attractive forces)
»z = 1, ideal behavior
»z < 1 attractive forces dominate, moderate
pressures
»z > 1 repulsive forces dominate, high
pressures

9. Real Gases - Other Equations of State
Virial Equation
Consider carbon dioxide
»At high temperatures (>50°C) and high
molar volumes (V
m > 0.3 L/mol),
isotherm looks close to ideal
»Suggests that behavior of real gases can
be approximated using a power series
(virial) expansion in n/V (1/V
m
)
{Kammerlingh-Onnes, 1911}
»Virial expansions common in physical
chemistry
CO
2
pV
mRT1
B
V
m

C
V
m
2
.......







10. Virial Equation (continued)
Coefficients experimentally determined (see Atkins, Table 1.3)
»3
rd
coefficient less impt than 2
nd
, etc.

B/V
m
>> C/V
m
2

For mixtures, coeff. depend on mole fractions
»B = x
1
2
B
11
+ 2 x
1
x
2
B
12
+ x
2
2
B
22
»x
1
x
2
B
12
represents interaction between gases
The compressibility factor, Z, is a function of p (see earlier figure) and T
»For ideal gas dZ/dp (slope of graph) = 0
Why?
»For real gas, dZ/dp can be determined using virial equation

Substitute for Vm (V
m = Z V
m°); and V
m°=RT/p
Slope = B’ + 2pC’+ ….

As p  0, dZ/dP  B’, not necessarily 0. Although eqn of state approaches ideal behavior
as p  0, not all properties of gases do
»Since Z is also function of T there is a temperature at which Z  1 with zero slope -
Boyle Temperature, T
B


At T
B , B’  0 and, since remaining terms in virial eqn are small, p V
m = RT for real gas
2
nd
Virial Coefficients
Equimolar Mixtures of CH
4 and CF
4
K B
1(CH
4)
(cm
3
/mol)
B
1(CH
4)
(cm
3
/mol)
B
12
(cm
3
/mol)
273.15 -53.35 -111.00 -62.07
298.15 -42.82 -88.30 -48.48
373.15 -21.00 -43.50 -20.43

11. Critical Constants
Consider what happens when you compress a real gas
at constant T (move to left from point A)
»Near A, P increases by Boyle’s Law
»From B to C deviate from Boyle’s Law, but p still
increases
»At C, pressure stops increasing
Liquid appears and two phases present (line CE)
Gas present at any point is the vapor pressure of the liquid
»At E all gas has condensed and now you have liquid
As you increase temperature for a real gas, the region
where condensation occurs gets smaller and smaller
At some temperature, T
c, only one phase exists across
the entire range of compression
»This point corresponds to a certain temperature, T
c
,
pressure, P
c , and molar volume, V
c , for the system
T
c, P
c , V
c are critical constants unique to gas
»Above critical point one phase exists (super critical
fluid), much denser than typical gases
CO
2
2 phases

12. Real Gases - Other Equations of State
Virial equation is phenomenolgical, i.e., constants depend on the
particular gas and must be determined experimentally
Other equations of state based on models for real gases as well as
cumulative data on gases
»Berthelot (1898)
Better than van der Waals at pressures not much above 1 atm
a is a constant
»van der Waals (1873)

»Dieterici (1899)
p
n
2
a
TV
2





(VnB)nRT
p
nRT
Vnb
a
n
V






2

RT
V
m
b

a
V
m
2
p
RTe
a/RTV
m
V
m
b

13. van der Waals Equation
Justification for van der Waals Equation
»Repulsion between molecules accounted for by
assuming their impenetrable spheres
Effective volume of container reduced by a
number proportional to the number of
molecules times a volume factor larger than
the volume of one molecule
Thus V becomes (V-nb)
b depends on the particular gas
He small, Xe large, b
Xe
>b
Ar
»Attractive forces act to reduce the pressure
Depends on both frequency and force of
collisions and proportional to the square of the
molar volume (n/V)
2
Thus p becomes p + a (n/V)
2
a depends on the particular gas
He inert, CO
2
less so, a
CO2
>>a
Ar
p
nRT
Vnb
a
n
V






2

RT
V
m
b

a
V
m
2
van der Waals Constants
gas a
(atm L
2
/ mol
2
)
b
(10
-2
L
2
/ mol)
Ar 1.337 3.20
CO2 3.610 4.29
He 0.0341 2.38
Xe 4.137 5.16

14. van der Waals Equation - Reliability
Above Tc, fit is good
Below Tc, deviations
Ideal vs van derWaals Isotherms @ n = 1
0
10
20
30
40
50
60
0.1 1 10
V (Liters)
Ideal Gas (150K)
Xe (150K)
Ideal Gas (450K)
Xe (450 K)
Ideal Gas (289.75K)
Xe (Tc=289.75K)

15. van der Waal’s Loops (cont.)
T = 260K; n = 0.25 moles
0
50
100
150
200
250
300
350
400
0.01 0.1 1
V (L)
Ideal Gas
CO2 (van der Waals)
CO
2 Critical Temperature 304.2 K (31.05°C)
Below Tc, oscillations occur
»van der Waals loops
»Unrealistic suggest that increase in p can increase V
»Replaced with straight lines of equal areas (Maxwell construction)

16. van der Waals Equation - Reliability
CO
2
van der Waals @T/Tc

17. van der Waals Equation
Effect of T and V
m
»Ideal gas isotherms obtained
»2nd term becomes negligible at high enough T
»1st term reduces to ideal gas law at high enough V
m
At or below T
c
»Liquids and gases co-exist
»Two terms come into balance in magnitude and oscillations occur
1st is repulsive term, 2nd attractive
»At T
c, we should have an flat inflexion point, i.e., both 1st and 2nd derivatives of equation w.r.t
V
m
= 0
p
nRT
Vnb
a
n
V






2

RT
V
m
b

a
V
m
2
dp
dV
m

d
RT
V
m
b

a
V
m
2






dV
m

RT
V
mb
2
2a
V
m
3
0
d
2
p
dV
m

d
RT
V
mb
2
2a
V
m
3








dV
m

2RT
V
m
b
3
6a
V
m
4
0
» Solving these equations for p,V
m
and T gives p
c
,V
c
and T
c
in terms
of a and b
Hint: you must use original eqn to do this
p
c
= a/27b
3
, V
c
p
c
= 3b


and T
c
= 8a/27Rb
Critical compression factor, Z
c, can be calculated using
definition for Z:
pVm = RTZ
Z
p
cV
c
RT
c

a
27b
2





3b
27Rb
R8a






3
8

18. Comparing Different Gases
Different gases have different
values of p, V and T at their
critical point
You can compare them at any
value by creating a reduced
variable by dividing by the
corresponding critical value
»p
reduced
= p
r
= p / p
c
; V
reduced
= V
r
=
V
m / V
c; T
reduced = T
r = T/ T
c
»This places all gases on the same
scale and they behave in a regular
fashion; gases at the same reduced
volume and temperature exert the
same reduced pressure.
Law of Corresponding States
Independent of equations of state
having two variables
p
r
p (atm)