SME
Thermal & Statistical Physics · Problem type
An Einstein solid: multiplicity and entropy
Use this for “An Einstein solid of 4 oscillators holds 5 units of energy. In how many ways can the energy be shared, and what is the entropy?”
Diagram
Row 6: 1, 6, 15, 20, 15, 6, 1; they add to 2⁶ = 64.
Entry 4 of row 6: C(6, 4) = 15.
The two above add to it: C(5, 3) + C(5, 4) = 10 + 5 = 15.
This is an example. Type your own values to start a new problem.
Ω
=
C(
q
+
N
−
1,
q
)
=
n
Places in the row
worked out (not typed)
σ
Entropy over kB
worked out (not typed)
S
Entropy
worked out (not typed) · J/K
Formulas
n
=
q
+
N
−
1
6 = 4 + 3 − 1
Ω
=
C(
n
,
q
)
15 = C(6, 4)
σ
=
ln(
Ω
)
2.7081 = ln(15)
S
=
1.381 ×
10
-23
×
σ
3.7398 ×
10
-23
=
1.381 ×
10
-23
×
2.7081
Assumptions
•
Each oscillator’s levels are evenly spaced by ħω, so the energy is q whole units, and the oscillators can be told apart.
•
Write the q units and N − 1 walls between oscillators in a row: Ω counts the ways to place the q units among the q + N − 1 spots.
•
Every microstate is equally likely, so S = kB ln Ω, with kB = 1.381 × 10⁻²³ J/K.
Step-by-step
WE KNOW
N = 3, q = 4
FIND
places in the row (n), multiplicity (Ω), entropy over kB (σ), entropy (S)
Step 1 · Find places in the row (n)
Use n = q + N − 1
The
q
units
and
the
N
−
1
walls
between
the
oscillators
fill
q
+
N
−
1
places
in
a
row.
n
=
4
+
3
−
1
n
=
6
Step 2 · Find multiplicity (Ω)
Use Ω = C(n, q)
Choose
which
q
of
the
places
hold
units:
entry
q
of
that
row
of
Pascal’s
triangle.
Ω
=
C(6,
4)
Ω
=
15
Step 3 · Find entropy over kB (σ)
Use σ = ln(Ω)
Boltzmann’s
entropy
in
units
of
kB,
σ
=
S
kB
:
the
log
of
the
number
of
microstates.
σ
=
ln(15)
σ
=
2.70805
Step 4 · Find entropy (S)
Use S = 1.381 × 10⁻²³ × σ
S
=
kBσ:
multiply
by
kB
for
joules
per
kelvin.
S
=
1.381 ×
10
-23
×
2.70805
S
=
3.7398 ×
10
-23
J/K
Check
Put the numbers back in. Both sides should match.
6 = 4 + 3 − 1 ✓
15 = C(6, 4) ✓
2.70805 = ln(15) ✓
3.7398 ×
10
-23
=
1.381 ×
10
-23
×
2.70805
✓
Topic check
See what you know: 5 questions on this topic
New questions each time. Pass to mark it complete.
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