SME
Chemical Engineering Thermodynamics · Problem type
Van der Waals pressure and volume
Use this for “CO₂ (Tc = 304.2 K, Pc = 73.83 bar) at 300 K has a molar volume of 1 L/mol. Find its pressure by van der Waals, and Z.”
Diagram
van
der
Waals
isotherm
of
the
gas
at
T
=
300
K:
P
=
RT
V − b
−
a
V
2
.
The
ideal
gas,
P
=
RT
V
,
is
dashed.
At
V
=
1
L/mol,
P
=
22.4
bar
against
24.94
bar
ideal:
attraction
(a)
pulls
it
down,
so
Z
=
PV
RT
=
0.898.
T
is
below
the
van
der
Waals
Tc
=
8a
27R
b
=
304.2
K,
so
the
isotherm
dips
and
rises
(where
the
real
gas
condenses);
far
out,
at
large
V,
it
meets
the
ideal
one.
This is an example. Type your own values to start a new problem.
a (Pa·m⁶/mol²)
=
27R²
×
(
Tc (K)
)
2
64
×
Pc (bar)
b (L/mol)
=
R
×
Tc (K)
8
×
Pc (bar)
P (bar)
=
R
×
T (K)
V (L/mol)
−
b (L/mol)
−
a (Pa·m⁶/mol²)
(
V (L/mol)
)
2
phase
Root kept
example
Z
Compressibility factor
calculated
Formulas
a
=
27
×
8.314
2
×
Tc
2
64 × Pc ×
10
5
0.3655
=
27
×
8.314
2
×
304.2
2
64 × 73.83 ×
10
5
b
=
8.314
×
Tc
8 × Pc ×
10
5
4.282 ×
10
-5
=
8.314
×
304.2
8 × 73.83 ×
10
5
P
=
(8.314
×
T
V
×
10
-3
−
b
−
a
÷
(
V
×
10
-3
)
2
)
÷
10
5
22.4
=
(8.314
×
300
1 ×
10
-3
− (4.282 ×
10
-5
)
−
0.3655
÷
(1 ×
10
-3
)
2
)
÷
10
5
Z
=
P
×
10
5
×
V
×
10
-3
8.314 ×
T
0.8982
=
22.4 ×
10
5
×
1 ×
10
-3
8.314 × 300
V
=
the
phase
root
of
P
×
V
3
−
(
P
×
b
×
10
3
+
0.08314
×
T
)
×
V
2
+
10
×
a
×
V
−
10
4
×
a
×
b
=
0
1
=
the
vapor
root
of
22.4
×
V
3
−
(22.4
×
(4.282
×
10
-5
)
×
10
3
+
0.08314
×
300)
×
V
2
+
10
×
0.3655
×
V
−
10
4
×
0.3655
×
(4.282
×
10
-5
)
=
0
Formulas are worked in m³/mol. The step-by-step shows the conversions.
Assumptions
•
The van der Waals equation, with a = 27R²Tc² ÷ (64Pc) and b = RTc ÷ (8Pc) from the critical point.
•
R = 8.314 J/(mol·K); a is in Pa·m⁶/mol², b and V are shown in L/mol and worked in m³/mol (SI).
•
Below Tc the cubic in V can have three roots: pick vapor for the greatest, liquid for the least.
Step-by-step
WE KNOW
Tc = 304.2 K, Pc = 73.83 bar, T = 300 K, V = 1 L/mol, phase: vapor
FIND
attraction constant (a), excluded volume (b), pressure (P), compressibility factor (Z)
Step 1 · Find attraction constant (a)
Use a = 27 × 8.314² × Tc² ÷ (64 × Pc × 10⁵)
From the critical point: square R and Tc, multiply by 27, divide by 64Pc (Pc in pascals).
a
=
27
×
8.314
2
×
304.2
2
64 × 73.83 ×
10
5
a
=
27
×
8.314
2
×
304.2
2
4.725 ×
10
8
a
=
27
×
69.12
×
92,537.64
4.725 ×
10
8
a
=
1,866.24
×
92,537.64
4.725 ×
10
8
a
=
1.727 ×
10
8
4.725 ×
10
8
a
=
0.3655
Pa·
m
6
/mol²
Step 2 · Find excluded volume (b)
Use b = 8.314 × Tc ÷ (8 × Pc × 10⁵)
The molecules’ own volume: RTc over 8Pc (Pc in pascals).
b
=
8.314
×
304.2
8 × 73.83 ×
10
5
b
=
8.314
×
304.2
5.9064 ×
10
7
b
=
2,529.1188
5.9064 ×
10
7
b
=
4.282 ×
10
-5
m
3
/mol
Step 3 · Find pressure (P)
Use P = (8.314 × T ÷ (V × 10⁻³ − b) − a ÷ (V × 10⁻³)²) ÷ 10⁵
V
in
m
3
/mol:
the
molecules’
volume
b
leaves
less
room
(
RT
V
−
b
),
their
pull
lowers
P
by
a
V
2
;
then
pascals
to
bar.
P
=
(8.314
×
300
1 ×
10
-3
− (4.282 ×
10
-5
)
−
0.3655
÷
(
10
-3
)
2
)
÷
10
5
P
=
(8.314
×
300
0.0009572
−
0.3655
÷
(
10
-3
)
2
)
÷
10
5
P
=
(8.314
×
300
÷
0.0009572
−
0.3655
÷
10
-6
)
÷
10
5
P
=
(2,494.2
÷
0.0009572
−
365,500)
÷
10
5
P
=
2,606,000 − 365,500
10
5
P
=
2,240,500
10
5
P
=
2,240,500
100,000
P
=
22.403
bar
Step 4 · Find compressibility factor (Z)
Use Z = P × 10⁵ × V × 10⁻³ ÷ (8.314 × T)
Compare
PV
with
RT,
both
in
SI:
Z
below
1
means
the
pull
between
molecules
wins.
Z
=
22.403 ×
10
5
×
10
-3
8.314 × 300
Z
=
(22.403
×
10
5
)
×
10
-3
2,494.2
Z
=
2,240.3
2,494.2
Z
=
0.8982
Step 5 · Convert the answers
Change each answer back to the unit you chose.
b = 4.282 × 10⁻⁵ m³/mol = 0.04282 L/mol (1 m³/mol = 1000 L/mol)
Check
Put the numbers back in. Both sides should match.
0.3655
=
27
×
8.314
2
×
304.2
2
64 × 73.83 ×
10
5
✓
4.282 ×
10
-5
=
8.314
×
304.2
8 × 73.83 ×
10
5
✓
22.403
=
(8.314
×
300
1 ×
10
-3
− (4.282 ×
10
-5
)
−
0.3655
÷
(
10
-3
)
2
)
÷
10
5
✓
0.8982
=
22.403 ×
10
5
×
10
-3
8.314 × 300
✓
1
=
the
vapor
root
of
22.403
×
V
3
−
(22.403
×
(4.282
×
10
-5
)
×
10
3
+
0.08314
×
300)
×
V
2
+
10
×
0.3655
×
V
−
10
4
×
0.3655
×
(4.282
×
10
-5
)
=
0
✓
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