SME
Thermal & Statistical Physics · Problem type
Thermal wavelength and an ideal gas’s Z₁
Use this for “Find the thermal wavelength of N₂ (28.01 u) at 77 K. Is nitrogen gas with 10²⁶ molecules per m³ still classical?”
Table
Temperature (T)
(K)
Thermal wavelength (λₜₕ)
(nm)
This is an example. Type your own values to start a new problem.
λ
=
h
÷
2π
m (u)
k
B
T (K)
=
λₜₕ (nm)
V
Volume
example · L
Z₁
One-particle partition function
worked out (not typed)
n
Number density
example · m⁻³
nλₜₕ³
Quantum overlap
worked out (not typed)
Formulas
λₜₕ
=
6.626 ×
10
-34
÷
√
2π ×
m
× 1.661 ×
10
-27
× 1.381 ×
10
-23
×
T
×
10
9
0.05036
=
6.626 ×
10
-34
÷
√
2π × 4.003 × 1.661 ×
10
-27
× 1.381 ×
10
-23
× 300
×
10
9
Z₁
=
V
×
10
-3
÷
(λₜₕ ×
10
-9
)
3
7.8275 ×
10
27
=
1 ×
10
-3
÷
(0.05036 ×
10
-9
)
3
nλₜₕ³
=
n
×
(λₜₕ ×
10
-9
)
3
3.1245 ×
10
-6
=
2.4457 ×
10
25
×
(0.05036 ×
10
-9
)
3
Assumptions
•
An ideal gas of point particles in a box of volume V; h = 6.626 × 10⁻³⁴ J·s and kB = 1.381 × 10⁻²³ J/K.
•
Mass in u (1.661 × 10⁻²⁷ kg): helium 4.003, N₂ 28.01, an electron 5.486 × 10⁻⁴.
•
One particle’s Z₁ = V ÷ λₜₕ³ counts the boxes of side λₜₕ that fit in V (its moving states).
•
Boltzmann counting holds while nλₜₕ³ ≪ 1; near 1, quantum statistics take over. For a gas, n = P ÷ kBT.
Step-by-step
WE KNOW
m = 4.003 u, T = 300 K, V = 1 L, n = 2.4457 × 10²⁵ m⁻³
FIND
thermal wavelength (λₜₕ), one-particle partition function (Z₁), quantum overlap (nλₜₕ³)
Step 1 · Find thermal wavelength (λₜₕ)
Use λₜₕ = 6.626 × 10⁻³⁴ ÷ √(2π × m × 1.661 × 10⁻²⁷ × 1.381 × 10⁻²³ × T) × 10⁹
Planck’s
constant
over
a
typical
thermal
momentum
√
2πmkBT
,
with
m
in
kg
(u
×
1.661 ×
10
-27
);
×
10
9
gives
nm.
λₜₕ
=
6.626 ×
10
-34
÷
√
2π × 4.003 × 1.661 ×
10
-27
× 1.381 ×
10
-23
× 300
×
10
9
λₜₕ = 0.0503647 nm
Step 2 · Find one-particle partition function (Z₁)
Use Z₁ = V × 10⁻³ ÷ (λₜₕ × 10⁻⁹)³
Summing
e
−E/kBT
over
the
moving
states
of
a
box
gives
V
λₜₕ
3
,
with
V
in
m
3
(L
×
10
-3
)
and
λₜₕ
in
m
.
Z₁
=
10
-3
÷
(0.0503647 ×
10
-9
)
3
Z₁
=
10
-3
1.2776 ×
10
-31
Z₁
=
7.8275 ×
10
27
Step 3 · Find quantum overlap (nλₜₕ³)
Use nλₜₕ³ = n × (λₜₕ × 10⁻⁹)³
The number of particles in one box of side λₜₕ: well below 1, their waves rarely overlap.
nλₜₕ³
=
2.4457 ×
10
25
×
(0.0503647 ×
10
-9
)
3
nλₜₕ³
=
(2.4457
×
10
25
)
×
(1.2776
×
10
-31
)
nλₜₕ³
=
3.1245 ×
10
-6
→
nλₜₕ³
<
0.01:
classical
(Boltzmann
counting
holds).
Check
Put the numbers back in. Both sides should match.
0.0503647
=
6.626 ×
10
-34
÷
√
2π × 4.003 × 1.661 ×
10
-27
× 1.381 ×
10
-23
× 300
×
10
9
✓
7.8275 ×
10
27
=
10
-3
÷
(0.0503647 ×
10
-9
)
3
✓
3.1245 ×
10
-6
=
2.4457 ×
10
25
×
(0.0503647 ×
10
-9
)
3
✓
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