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Modern Engineering Thermodynamics

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12.2 Thermodynamic Properties of Gas Mixtures 409<br />

Table 12.3 Total, Molar Specific, and Partial Molar Specific Properties<br />

of a Mixture of Gases<br />

Total Property<br />

of the Mixture<br />

V m = ∑ N<br />

n i^v i<br />

i=1<br />

U m = ∑ N<br />

n i^u i<br />

i=1<br />

H m = ∑ N<br />

i=1<br />

n i^hi<br />

S m = ∑ N<br />

n i^s i<br />

i=1<br />

Molar Specific Property<br />

of the Mixture<br />

v m = V m /n m = ∑ N<br />

u m = U m /n m = ∑ N<br />

h m = H m /n m = ∑ N<br />

s m = S m /n m = ∑ N<br />

ni<br />

nm<br />

i=1<br />

ni<br />

nm<br />

i=1<br />

ni<br />

nm<br />

i=1<br />

ni<br />

nm<br />

i=1<br />

^v i = ∑ N<br />

i=1<br />

^u i = ∑ N<br />

i=1<br />

^<br />

h i = ∑ N<br />

^s i = ∑ N<br />

i=1<br />

i=1<br />

χ<br />

^v i = ∂V m<br />

i^v i ∂n i<br />

<br />

χ<br />

^u i = ∂U <br />

m<br />

i^u i ∂n i<br />

<br />

^hi<br />

χ<br />

= ∂H <br />

m<br />

i^hi ∂n i<br />

<br />

χ<br />

^s i = ∂S <br />

m<br />

i^s i ∂n i<br />

Partial Molar Specific Property<br />

of Gas i in the Mixture<br />

<br />

pm,Tm,nj<br />

pm,Tm,nj<br />

pm,Tm,nj<br />

pm,Tm,nj<br />

Table 12.4 Specific Heats of a Mixture and the Partial Specific Heats<br />

of the Gases in the Mixture<br />

Specific Heat of the Mixture<br />

Partial Specific Heat of Gas i in the Mixture<br />

<br />

c vm = ∂u <br />

<br />

m<br />

=<br />

∂T m<br />

∑ N<br />

w<br />

^c i^c vi = ∂^u <br />

i<br />

vi ∂T m vm<br />

vm i=1<br />

<br />

c pm = ∂h <br />

<br />

m<br />

=<br />

∂T m<br />

∑ N<br />

w<br />

^c i^c pi = ∂^h <br />

i<br />

pi ∂T m pm<br />

vm<br />

<br />

c vm = ∂u <br />

m<br />

∂T m<br />

<br />

c pm = ∂h <br />

m<br />

∂T m<br />

vm<br />

vm<br />

i=1<br />

= ∑ N<br />

i=1<br />

= ∑ N<br />

i=1<br />

χ<br />

^c vi = ∂^u i<br />

i^c vi ∂T m vm<br />

!<br />

χ i^c pi<br />

^c pi = ∂^h i<br />

∂T m<br />

<br />

<br />

vm<br />

therefore, when either w i , χ i , ψ i ,orπ i is multiplied by 100, it represents the composition percentage of gas<br />

i present on a mass, molar, volume, or pressure basis, respectively. Note, however, that w i and χ i do not have<br />

the same numerical values; therefore, a mass based percentage analysis of the composition depends on which fractional<br />

base is used in its determination.<br />

If we consider the mixture to be a unique substance, then we can compute its equivalent molecular mass M m from<br />

Eqs. (12.1), (1.9), and (12.4) as<br />

M m = m m<br />

n m<br />

= 1<br />

n m<br />

∑ N !<br />

m i = ∑ N<br />

n i M i<br />

n<br />

=<br />

m<br />

∑ N<br />

χ i M i (12.11)<br />

i=1 i=1 i=1<br />

and using Eqs. (12.2), (1.9), and (12.3) as<br />

M m = m m<br />

n m<br />

=<br />

m m<br />

∑ N<br />

m i<br />

i=1<br />

M i<br />

! =<br />

∑ N<br />

i=1<br />

1<br />

! =<br />

ðm i /m m Þ<br />

M i<br />

1<br />

∑ N<br />

w i<br />

i=1<br />

M i<br />

! (12.12)<br />

Using Eqs. (12.11) and (12.12), we can now easily convert back and forth between mass and mole<br />

fractions with<br />

w i = m i<br />

m m<br />

=<br />

n iM i<br />

n m M m<br />

= χ i<br />

<br />

M i<br />

M m<br />

(12.13)

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