SME
Thermal & Statistical Physics · Problem type
Maxwell’s speeds: most probable, mean and rms
Use this for “Find the most probable, mean and rms speeds of oxygen molecules (32.00 g/mol) at 250 K.”
Chart
Most probable vₚ = 422 m/s < mean ⟨v⟩ = 476.2 m/s < rms vᵣₘₛ = 516.9 m/s; the area under the curve is 1
This is an example. Type your own values to start a new problem.
v
p
=
2R
T (K)
÷
M (g/mol)
=
vₚ (m/s)
⟨v⟩
Mean speed
calculated · m/s
vᵣₘₛ
Root-mean-square speed
calculated · m/s
Formulas
vₚ
=
√(2
×
8.314
×
T
M
÷ 1,000
)
422
=
√(2
×
8.314
×
300
28.01 ÷ 1,000
)
⟨v⟩
=
√(8
×
8.314
×
T
π ×
M
÷ 1,000
)
476.2
=
√(8
×
8.314
×
300
π × 28.01 ÷ 1,000
)
vᵣₘₛ
=
√(3
×
8.314
×
T
M
÷ 1,000
)
516.9
=
√(3
×
8.314
×
300
28.01 ÷ 1,000
)
Assumptions
•
An ideal gas in equilibrium at T: its speeds follow Maxwell’s distribution, f(v) ∝ v²e−mv²/2kBT.
•
Per mole, kB ÷ m = R ÷ M with R = 8.314 J/(mol·K); M in g/mol is divided by 1000 for kg/mol.
•
Always vₚ < ⟨v⟩ < vᵣₘₛ (as √2, √(8/π) and √3): the tail of fast molecules pulls the mean and rms up.
Step-by-step
WE KNOW
M = 28.01 g/mol, T = 300 K
FIND
most probable speed (vₚ), mean speed (⟨v⟩), root-mean-square speed (vᵣₘₛ)
Step 1 · Find most probable speed (vₚ)
Use vₚ = √(2 × 8.314 × T ÷ (M ÷ 1,000))
The
peak
of
f
(v):
set
df/dv
=
0,
and
v
2
=
2kBT
÷
m
=
2R
T
M
.
vₚ
=
√(2
×
8.314
×
300
28.01 ÷ 1,000
)
vₚ
=
√(2
×
8.314
×
300
0.02801
)
vₚ
=
√(16.628
×
300
0.02801
)
vₚ
=
√(
4,988.4
0.02801
)
vₚ
=
√
178
,100
vₚ = 422 m/s
Step 2 · Find mean speed (⟨v⟩)
Use ⟨v⟩ = √(8 × 8.314 × T ÷ (π × M ÷ 1,000))
The
average
of
v
over
f
(v),
∫v
f
(v)
dv
=
√
8kBT ÷ πm
,
written
per
mole.
⟨v⟩
=
√(8
×
8.314
×
300
π × 28.01 ÷ 1,000
)
⟨v⟩
=
√(8
×
8.314
×
300
0.088
)
⟨v⟩
=
√(66.512
×
300
0.088
)
⟨v⟩
=
√(
19,953.6
0.088
)
⟨v⟩
=
√
226
,700
⟨v⟩ = 476.2 m/s
Step 3 · Find root-mean-square speed (vᵣₘₛ)
Use vᵣₘₛ = √(3 × 8.314 × T ÷ (M ÷ 1,000))
Each
molecule
averages
1
2
m⟨
v
2
⟩
=
(
3
2
)kBT,
so
⟨
v
2
⟩
=
3R
T
M
;
vᵣₘₛ
is
its
square
root.
vᵣₘₛ
=
√(3
×
8.314
×
300
28.01 ÷ 1,000
)
vᵣₘₛ
=
√(3
×
8.314
×
300
0.02801
)
vᵣₘₛ
=
√(24.942
×
300
0.02801
)
vᵣₘₛ
=
√(
7,482.6
0.02801
)
vᵣₘₛ
=
√
267
,100
vᵣₘₛ = 516.9 m/s
Check
Put the numbers back in. Both sides should match.
422
=
√(2
×
8.314
×
300
28.01 ÷ 1,000
)
✓
476.2
=
√(8
×
8.314
×
300
π × 28.01 ÷ 1,000
)
✓
516.9
=
√(3
×
8.314
×
300
28.01 ÷ 1,000
)
✓
Topic check
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