SME
Thermal & Statistical Physics
Laws of thermodynamics
Use this for “2.00 mol of an ideal gas expands slowly at 400 K from 5 L to 15 L. Find the work it does, the heat it takes in and its entropy change.”
Diagram
PV = nRT = 2,494 kPa·L at both ends: P₁ = 249 kPa, P₂ = 125 kPa.
W = nRT ln(V₂/V₁)
W
=
2,494
×
ln(
20
10
)
=
+1,729
J
The shaded area is 1,729 J.
The gas expands: it does work, so W is positive.
Q = +1,729 J.
This is an example. Type your own values to start a new problem.
W
=
n (mol)
R
T (K)
ln
V₂ (L)
V₁ (L)
=
W (J)
Q
Heat taken in
calculated · J
ΔS
Entropy change of the gas
calculated · J/K
Formulas
W
=
n
×
8.314
×
T
×
ln(
V₂
V₁
)
+1,729
=
1
×
8.314
×
300
×
ln(
20
10
)
Q
=
W
+1,729 = 1,729
ΔS
=
Q
T
+5.763
=
1,729
300
Assumptions
•
An ideal gas changed slowly (quasi-static) at one temperature, so ΔU = 0 and the first law ΔU = Q − W gives Q = W.
•
W is the work done by the gas, the area under the P–V curve: ∫P dV with P = nRT ÷ V; R = 8.314 J/(mol·K).
•
The heat comes in at one temperature, so ΔS = Q ÷ T, in kelvins.
•
ΔS depends only on the start and end: a free expansion between the same volumes has the same ΔS, though Q = W = 0.
Step-by-step
WE KNOW
n = 1 mol, T = 300 K, V₁ = 10 L, V₂ = 20 L
FIND
work done by the gas (W), heat taken in (Q), entropy change of the gas (ΔS)
Step 1 · Find work done by the gas (W)
Use W = n × 8.314 × T × ln(V₂ ÷ V₁)
Add
up
P
dV
with
P
=
nRT
V
from
V₁
to
V₂
:
nRT
comes
out
front
and
∫
dV
V
is
ln(
V₂
V₁
).
W
=
1
×
8.314
×
300
×
ln(
20
10
)
W
=
8.314
×
300
×
ln(2)
W
=
8.314
×
300
×
0.6931
W
=
2,494.2
×
0.6931
W
=
+1,729
J
→
W
>
0:
the
gas
expands
and
pushes
the
piston
out.
Step 2 · Find heat taken in (Q)
Use Q = W
ΔU
=
0
at
one
temperature,
so
the
first
law
ΔU
=
Q
−
W
makes
the
heat
in
equal
the
work
out.
Q
=
+1,729
Q
=
+1,729
J
Step 3 · Find entropy change of the gas (ΔS)
Use ΔS = Q ÷ T
The
heat
comes
in
slowly
at
one
temperature,
so
divide
it
by
T
.
ΔS
=
1,729
300
ΔS = +5.763 J/K
Check
Put the numbers back in. Both sides should match.
+1,728.8477
=
1
×
8.314
×
300
×
ln(
20
10
)
✓
+1,728.8477 = 1,728.8477 ✓
+5.7628
=
1,728.8477
300
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
More problem types
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