Partial volumes of all species equal to total volume

[Floris van de Beek]

Hi all,

I'm trying to retrieve the partial (molar) volumes of the species in my reactor, but they all have the same value. Now I'm not sure if I'm misunderstanding the concept of volume fractions or that I've run into a bug, so I'd like to get some input. Here's a minimum working example:

import cantera as ct

gas = ct.Solution('gri30.yaml')

gas.TPX = 300, ct.one_atm, 'CH4:1.0'

q1 = ct.Quantity(gas, mass=1) # 1 kg of methane

print ('Quantity 1 volume: {} and moles: {}'.format(q1.volume, q1.moles))

gas.X = 'O2:1.0'

q2 = ct.Quantity(gas, mass=1) # 1 kg of oxygen

print ('Quantity 2 volume: {} and moles: {}'.format(q2.volume, q2.moles))

q3 = q1 + q2

print ('Quantity 3 volume: {} and moles: {}'.format(q3.volume, q3.moles))

print ('Quantity 3 partial volumes: {}'.format(q3.partial_molar_volumes * q3.moles))

This is the terminal output:

Quantity 1 volume: 1.5344517748730229 and moles: 0.06233248145608677
Quantity 2 volume: 1.5344517748730229 and moles: 0.03125195324707794
Quantity 3 volume: 3.0689035497460457 and moles: 0.09358443470316472
Quantity 3 partial volumes: [3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355
 3.06890355 3.06890355 3.06890355 3.06890355 3.06890355]

I was expecting that the sum of all partial volumes would add up to the total volume (3.0689 m3), instead of each indivual partial volume being equal to the total volume.

Please advice, thanks!

Floris

[Ray Speth]

Hi Floris,

All of the partial molar properties should be thought of as being “per mole of species k”, not “per mole of mixture”. Therefore, for an ideal gas, Cantera defines the partial molar volume for all species to be , as described in the documentation. This means that the total volume can be computed as , the same way the molar enthalpy can be computed as .

Regards,
Ray

​

[William R. Smith]

Hi Floris (and all "thermodynamic nerds" out there):

Partial molar properties are only defined at a given (T,P), and along with all other extensive properties at (T,P), (such as enthalpy and Gibbs energy), the total property value of a solution  phase can be expressed in terms of the partial molar properties, by means of the equation  

Vt(T,P)=sum n_k* V(pm)_k(T,P,x)  (1)

where Vt is the total volume of the system, n_k is the molar amount of species k,  the partial molar volume V(pm)_k(T,P,x) is the partial molar volume of species k, defined by (dV/dnk) at constant (T,P), and x denotes the composition, for example, by means of mole fractions. (Note also that dividing by the total number of moles nt Eq. (1) expresses the molar volume V/nt in terms of the mole fractions and the V(pm) quantities.)

Equation (1) is ALWAYS true, both for real(nonideal) and ideal gas mixtures  (it's also true for liquid and solid solutions as well).  It's a mathematical consequence of the fact that V(T,P) is an "extensive quantity" (technically, this means that it's a homogeneous function of degree 1 in the molar amounts).

It's interesting to note there is an approximation for real gases called "Amagat's Law", which reads similarly to Eq. (1):

V(T,P)=sum n_k* V(pc)_k (T,P) (2)

where V(pc)_k is the "pure component volume" of species k, whihch is the volume of the pure component k at the (T,P) of the mixture.  

There is often confusion about the meaning of "Amagat's Law", since it's not a "law" in the proper sense, but an "approximation" for real gases.  This confusion arises when it is only applied in the context of an ideal gas mixture.  In that case it is exact, since it is simply a property of ideal gases: To see this, applying Eq. (2) to an ideal gas mixture gives

V(pc)_k = n_k RT/P, 

which gives

sum n_k* V(pm)_k(T,P,x) = sum n_k RT/P = RT/P sum n_k = n_t RT/P = Vt

where n_t is the total number of moles, and v_t is the total volume.

(Similar confusion arises about the meaning of "Dalton's Law of Partial Pressures", which like Amagat's Law is an approximation rather than a law.  See, for example the paper Missen, R. W. and Smith, W. R.,  "A note on Dalton's law: Myths, facts, and implementation", J. Chem. Educ.,  82{8}, 1197-1201, 2005)