SME
Chemical Engineering Thermodynamics · Problem type
Margules activity coefficients
Use this for “A binary follows the one-parameter Margules equation with A = 0.8. Find γ₁ and γ₂ at x₁ = 0.3.”
Diagram
Pxy diagram of 1–2: the bubble curve runs from P₂ˢᵃᵗ = 406.7 to P₁ˢᵃᵗ = 1,021 mmHg, the dew curve under it; between them liquid and vapor coexist.
Margules A = 0.8: ln γ₁ = A x₂², ln γ₂ = A x₁², so γ₁ = 1.48 and γ₂ = 1.07 at x₁ = 0.3.
The curves bulge above Raoult’s straight line (positive deviation).
P = x₁γ₁P₁ˢᵃᵗ + x₂γ₂P₂ˢᵃᵗ = 759.2 mmHg, y₁ = 0.597 at x₁ = 0.3.
This is an example. Type your own values to start a new problem.
ln
γ₁
=
A
(1
−
)
2
ln
γ₂
=
A
×
x₁
2
With x₂ = 1 − x₁, the first line is the picture’s ln γ₁ = A x₂².
Formulas
ln(
γ₁
)
=
A
×
(1 −
x₁
)
2
ln(1.48)
=
0.8
×
(1 − 0.3)
2
ln(
γ₂
)
=
A
×
x₁
2
ln(1.075)
=
0.8
×
0.3
2
Assumptions
•
One-parameter Margules: the excess Gibbs energy is A x₁x₂RT, so ln γ₁ = A x₂² and ln γ₂ = A x₁².
•
Modified Raoult’s law, yᵢP = xᵢγᵢPᵢsat; A above 0 raises the curves over Raoult’s dashed line.
•
The picture uses 1021 and 406.7 mmHg (benzene and toluene at 90 °C) to show the shape.
•
Above A = 2 the liquid splits into two phases, and these γ no longer describe one liquid.
Step-by-step
WE KNOW
A = 0.8, x₁ = 0.3
FIND
activity coefficient of 1 (γ₁), activity coefficient of 2 (γ₂)
Step 1 · Find activity coefficient of 1 (γ₁)
Use ln(γ₁) = A × (1 − x₁)²
Square
the
other
liquid’s
mole
fraction
x₂
=
1
−
x₁
,
multiply
by
A
,
then
raise
e
to
it.
γ₁
=
e^(
A
×
(1 −
x₁
)
2
)
γ₁
=
e^(0.8
×
(1 − 0.3)
2
)
γ₁
=
e
0.8 ×
0.7
2
γ₁
=
e
0.8 × 0.49
γ₁
=
e
0.392
γ₁
=
1.48
Step 2 · Find activity coefficient of 2 (γ₂)
Use ln(γ₂) = A × x₁²
Square
x₁
,
multiply
by
A
,
then
raise
e
to
it.
γ₂
=
e
A
×
x₁
2
γ₂
=
e
0.8 ×
0.3
2
γ₂
=
e
0.8 × 0.09
γ₂
=
e
0.072
γ₂
=
1.075
Check
Put the numbers back in. Both sides should match.
ln(1.48)
=
0.8
×
(1 − 0.3)
2
✓
ln(1.0747)
=
0.8
×
0.3
2
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
Related lessons
Vapor–liquid equilibrium
Main lesson
Dew pressure
Use this for “A vapor of 50% benzene and 50% toluene is at 90 °C, where P₁sat = 1021 mmHg and P₂sat = 406.7 mmHg. Find its dew pressure.”
Binary flash
Use this for “A feed of 60% benzene and 40% toluene flashes at 90 °C and 760 mmHg (P₁sat = 1021 mmHg, P₂sat = 406.7 mmHg). Find x₁, y₁ and V ÷ F.”