SME
University Physics I: Mechanics
First and second laws of thermodynamics
Use this for “One mole of ideal gas at 300 K doubles its volume isothermally. Find W, Q and ΔS.”
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 (J)
=
n (mol)
×
8.314
×
T (K)
×
ln
V₂ (L)
V₁ (L)
Q
Heat into the gas
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 a fixed temperature, so its internal energy stays the same: ΔU = 0.
•
W is the work done by the gas: above 0 when it expands, below 0 when it is squeezed; R = 8.314 J/(mol·K).
•
The first law ΔU = Q − W with ΔU = 0 gives Q = W; the heat enters at one temperature, so ΔS = Q ÷ T.
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 into the gas (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)
W
is
the
area
under
the
P–V
curve,
∫P
dV
with
P
=
nRT
V
;
nRT
is
fixed,
and
∫
dV
V
from
V₁
to
V₂
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,
pushing
the
piston
out,
and
takes
in
as
much
heat.
Step 2 · Find heat into the gas (Q)
Use Q = W
The
temperature
does
not
change,
so
ΔU
=
0,
and
the
first
law
ΔU
=
Q
−
W
leaves
Q
=
W
:
every
joule
of
heat
in
leaves
as
work.
Q
=
+1,729
Q
=
+1,729
J
Step 3 · Find entropy change of the gas (ΔS)
Use ΔS = Q ÷ T
The
heat
flows
in
reversibly
at
one
temperature,
so
∫
dQ
T
is
just
Q
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
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