Corresponding State

69
Principle of Corresponding States

___________________________________________________________________________________________ _
In many areas of science it is useful to compare a fundamental property of different objects on a relative scale for purposes of comparison. For different gases, the reduced variables are used Reduced variables
Reduced Pressure
Pr = P Pc
Reduced Temperature
Tr = T Tc
Reduced Volume
V Vmr = m Vmc
Pc, Tc, Vmc critical constants
70
Law of corresponding states: * All gases will correspond to the same equation of state expressed in reduced variables * Real gases at similar reduced volume and temperature will produce the same reduced pressure. If two gases have the same value of Tr, Pr, Vmr, they are in corresponding states. Example 1.7 He, Vm = 10.5 L mol-1, P = 0.1 atm, T=10 K. Determine Vm, P, T of CO2 at the corresponding state with He
Solution
Tc / K He CO2 5.2 304.2
Vmc / cm3 mol-1 57.8 94
Pc / atm 2.26 72.9
_______________________________________________________________________ _
71
For He:
Vmr = Vm 10.51000 cm3 mol -1 = =181.66 Vmc 57.8 cm3 mol -1
Tr = T = 10K =1.02 Tc 5.2K 1 Pr = P = 0. atm = 0.044 Pc 2.26 atm
CO2 at corresponding state: P = Pr x Pc = 0.044 x 72.9 atm = 3.2 atm
Vm = Vmr x Vmc = 1.81.66 x 94 cm3 mol-1 = 17.1 cm3 mol-1 T = Tr x Tc = 1.92 x 304.2 K = 584 K
72
Conclusion: He at 0.1 atm and 10 K in a molar volume of 10.5 L mol-1 and CO2 at 3.2 atm and 584 K in a volume of 17.1 cm3 mol-1 are in the same corresponding state. The compression factor is a convenient way of quantifying deviation from the ideal gas law. In calculating Z using the van der Waals equation of state, different parameter must be used for each gas. Van der Waals, who first tried this procedure, hoped that gases confined to the same reduced volume, Vmr, at the same reduced temperature, Tr, would exert the same reduced pressure, Pr. The hope was largely fulfilled (Figure 1.13). The illustration shows the dependence of the Z on the Pc for a variety of gases at various Tc. When this graph is compared with Fig.1.9,where similar data are plotted without using reduced variables. The observation that real gases at the same Vmc and Tr exert the same Pr.
73
Values for the compression factors, Z, shown as a function of the reduced pressure, Pr, for four gases at three values of reduced temperature, Tr - The law of corresponding states can be used by many gases
-
Parameters a and b can be eliminated from the vdW equation of state by expressing the equation in terms reduced variables Tr, Pr and Vmr
V = VmrVmc , T = TrTc
P = PrPc ,
74
R TrTc a Pr Pc = Vmr Vmc b V 2mr V 2mc
Replace Tc, Vmc, and Pc by the relationships derived in equation (1.22)

Pc = a , 27 b2 Vmc = 3b, and Tc = 8a 27 R b
Equation (1.30) becomes

a Pr 8 a Tr a = 27 b ( 3 b Vmr b) 27 b2 9 b2 Vmr 2 8 Tr 3 Pr = 3 Vmr 1 Vmr 2
-
or
(1.31)
Equation (1.31) relates Tr, Pr and Vmr without reference to parameter a and b. - It has the character of a universal equation.
75
If the isotherms are plotted in terms of reduced variables, then the same curves are obtained whatever the gas is. - This is precisely the content of the principle of corresponding states, so the vdW equation is compatible with it.
-
Example 1.8 Using Figure 1.14 and the data given: Calculate the volume occupied by 1.0 kg of CH4 gas at T = 230 K and P = 68.0 bar. Calculate V Videal and the relative error in V if V were calculated from the ideal gas equation of state. Given: Tc/K = 190.56, Pc / bar = 45.99
76
Compression factor, Z, as a function of Pr for the Tr value indicated. The curves were calculated using the vdW equation of state From the data given, Tr and Pr can be calculated:
Tr = 230 K =1.21 and Pr = 68.0 bar =1.48 190.56 K 45.99 bar
From Figure 1.14, Z = 0.63

Z= PVm PV = RT nRT V = ZnRT P
77
0.63 V=
1000 g 0.08314 L bar K 1 mol 1 230 K 16.04 g mol 1 68. 0bar
Vm V Z= o= o Vm V V o = V = 11.04 L =17.52 L Z 0.63
= 11.04 L
Therefore V Videal = V Vo = 11.04 L 17.52 L = -6.48 L

V Videal Error = 100 % = 6.48 L 100 % =58.7 % V 11.04 L
7PCS6774

Corresponding State

Uploaded by

Copyright:

Available Formats

Corresponding State

Uploaded by

Document Information

Original Description:

Copyright

Available Formats

Share this document

Share or Embed Document

Sharing Options

Did you find this document useful?

Is this content inappropriate?

Copyright:

Available Formats

Corresponding State

Uploaded by

Copyright:

Available Formats

69

Principle of Corresponding States

Pc, Tc, Vmc critical constants

Tc / K He CO2 5.2 304.2

Vmc / cm3 mol-1 57.8 94

Pc / atm 2.26 72.9

Tr = T = 10K =1.02 Tc 5.2K 1 Pr = P = 0. atm = 0.044 Pc 2.26 atm

CO2 at corresponding state: P = Pr x Pc = 0.044 x 72.9 atm = 3.2 atm

R TrTc a Pr Pc = Vmr Vmc b V 2mr V 2mc

Replace Tc, Vmc, and Pc by the relationships derived in equation (1.22)

Equation (1.30) becomes

From Figure 1.14, Z = 0.63

1000 g 0.08314 L bar K 1 mol 1 230 K 16.04 g mol 1 68. 0bar

Vm V Z= o= o Vm V V o = V = 11.04 L =17.52 L Z 0.63

Therefore V Videal = V Vo = 11.04 L 17.52 L = -6.48 L

You might also like