P-v-T Relations for Vapors (Gases)
Consider a fixed amount (# of moles) of gas in a pistoncylinder assembly held at constant temperature, say T1.
Move piston slowly to various positions, measure system
pressure P and volume (via height h) at each equilibrium
position
T1
T1
P2
P1
h3, h4,...
h2
h1
Repeat tests at different temperatures T2, T3,…
volume
Define molar specific volume v as
# moles
Plot results as
 m3 


 mol 
Pv
vs P
T
Pv
T
x
o
$
#
x
o
$ - T3
$
$
#
o
x - T1
o - T2
# - T4
#
$
#
#
P
54
Since Pv is a finite number, as P  0, v  
As v (volume/#moles) approaches infinity continuum
assumption fails have to extrapolate experimental data to
P= 0
x
Pv
T
x
x
x
x
x
x
x
x
x
T1
T2
T3
T4
x
x
x
P
Extrapolated data
It is found that when the data is extrapolated to P= 0 you
get a limiting value for Pv / T :
N  m3 


2 
m  kmol 
Nm
 Pv 
lim   R
units :

o
P  0 T 
K
kmol oK
The same value for R is found for all gases, hence it is
referred to as the Universal Gas Constant
R  8314
J
kJ

8
.
314
kmol oK
kmol oK
55
The ratio of the molar and mass specific volume defines
the Molar mass, :
v  vol  mass mass




v  mol  vol
mol
v   v
 Pv 
 Pv 
lim   lim
R
P  0 T 
P  0 T 
 Pv  R
lim  
P  0 T 

Define the gas constant as, R  R / 
The gas constant for air is
Rair 
J  kmol 
J

  8314

286
.
7



 
kmol oK  29kg 
kg oK
R
We define the compressibility factor Z as
Z
Pv P ( v) Pv


R T ( R )T RT
56
Re-plotting data as Z vs P (as P 0,
Z
1
x
x
x
x
x
x
x
x
x
x
Pv
 R , Z 1)
T
T1
T2
T3
T4
x
x
x
P
Repeat experiments using different gases and get the same
shape curves.
If we plot Z versus the reduced pressure PR (=P/Pc) for
different reduced temperatures TR (=T/Tc) you get a
universal set of curves known as the generalized
compressibility chart
Molar mass and critical properties are given in Table A-1,
for example:
Tc(air)= 133K
Pc(air)= 37.7 bar
Tc(water)= 647.3K
Pc(water)= 220.9 bar
57
For a given gas with a gas constant R= R /  , knowing the
pressure and temperature you can use the above graph to
v
get the specific volume v 

Z 1 (within 5%) for low reduced pressure (PR < 0.2)
independent of temperature, e.g. Pair < 7.6 bar
Z 1 (within 5%) for high reduced temperatures (TR > 2)
independent of pressure, e.g. for air Tair > 266K (-7C)
58
In most engineering applications we deal with gases at
conditions that are within these limits, so Z1
A gas for which Z can be assumed equal to one is called a
perfect gas, or an ideal gas
Pv Pv
For an ideal gas Z 

1
R T RT
Pv  R T
and
since v  V/M
so
Pv  RT  P  RT
and
v  V/n
PV  nR T and
n  # moles
PV  MRT
Microscopic Point of View
Molecules have KE associated with motion and PE
associated with mutual attraction/repulsion. An ideal gas
is one where the PE is negligible, e.g., billiard ball model.
At low pressure molecules are far apart so negligible PE
At high temp KE >> PE
59