SME
Materials Science & Material Properties · Problem type
Vacancies at a temperature
Use this for “Copper has 8 × 10²⁸ atom sites per m³ and Qᵥ = 0.9 eV. How many vacancies per m³ are there at 1000 °C?”
Chart
Nᵥ(1,273) = 2.1871 × 10²⁵
This is an example. Type your own values to start a new problem.
Nᵥ (m⁻³)
=
N (m⁻³)
×
e
−
eV
÷
(0.00008617
×
K
)
Formulas
Nᵥ
=
N
×
e
−Qᵥ ÷ (0.00008617 ×
T
)
2.1871 ×
10
25
=
8 ×
10
28
×
e
−0.9 ÷ (0.00008617 × 1,273)
Assumptions
•
N counts the atom sites in each cubic meter: density × Avogadro’s number ÷ atomic mass.
•
Qᵥ is the energy to make one vacancy, in eV; k = 8.617 × 10⁻⁵ eV/K is Boltzmann’s constant.
•
T is in kelvins (°C + 273): 1000 °C is 1273 K.
Step-by-step
WE KNOW
N = 8 × 10²⁸ m⁻³, Qᵥ = 0.9 eV, T = 1,273 K
FIND
vacancies per m³ (Nᵥ)
Step 1 · Find vacancies per m³ (Nᵥ)
Use Nᵥ = N × e−Qᵥ ÷ (0.00008617 × T)
The
share
of
sites
left
empty
is
e
−Qᵥ ÷ kT
.
Scale
the
sites
by
that
share.
Nᵥ
=
8 ×
10
28
×
e
−0.9 ÷ (0.00008617 × 1,273)
Nᵥ
=
(8
×
10
28
)
×
e
−0.9 ÷ 0.1096944
Nᵥ
=
(8
×
10
28
)
×
e
−8.2046
Nᵥ
=
(8
×
10
28
)
×
0.00027339
Nᵥ
=
2.1871 ×
10
25
m
-3
Check
Put the numbers back in. Both sides should match.
2.1871 ×
10
25
=
8 ×
10
28
×
e
−0.9 ÷ (0.00008617 × 1,273)
✓
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Diffusion coefficient at a temperature
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Steady diffusion flux (Fick’s first law)
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