SME
Physical Chemistry I: Thermodynamics & Kinetics
Kinetic theory of gases
Use this for “Find the most probable, mean and root-mean-square speeds of CO₂ molecules (44.01 g/mol) at 350 K.”
Chart
Most probable vₚ = 420.7 m/s < mean ⟨v⟩ = 474.7 m/s < rms vᵣₘₛ = 515.3 m/s; the area under the curve is 1
This is an example. Type your own values to start a new problem.
vₚ (m/s)
=
2R
×
T (K)
÷
M (g/mol)
⟨v⟩ (m/s)
=
8R
×
T (K)
÷
(π
×
M (g/mol)
)
vᵣₘₛ (m/s)
=
3R
×
T (K)
÷
M (g/mol)
Formulas
vₚ
=
√(2
×
8.314
×
T
M
÷ 1,000
)
420.7
=
√(2
×
8.314
×
298.15
28.01 ÷ 1,000
)
⟨v⟩
=
√(8
×
8.314
×
T
π ×
M
÷ 1,000
)
474.7
=
√(8
×
8.314
×
298.15
π × 28.01 ÷ 1,000
)
vᵣₘₛ
=
√(3
×
8.314
×
T
M
÷ 1,000
)
515.3
=
√(3
×
8.314
×
298.15
28.01 ÷ 1,000
)
Assumptions
•
An ideal gas at equilibrium: the speeds of its molecules follow the Maxwell distribution, f(v) ∝ v²e−Mv²/2RT.
•
R = 8.314 J/(mol·K), and a joule is kg·m²/s², so the molar mass is turned into kg/mol for speeds in m/s.
•
The three speeds always come in the order vₚ < ⟨v⟩ < vᵣₘₛ, as √2, √(8/π) and √3 times √(RT ÷ M), whatever the gas.
Step-by-step
WE KNOW
M = 28.01 g/mol, T = 298.15 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
the
distribution:
setting
the
slope
of
f
(v)
to
zero
gives
v
2
=
2R
T
M
.
vₚ
=
√(2
×
8.314
×
298.15
28.01 ÷ 1,000
)
vₚ
=
√(2
×
8.314
×
298.15
0.02801
)
vₚ
=
√(16.628
×
298.15
0.02801
)
vₚ
=
√(
4,957.6382
0.02801
)
vₚ
=
√
177
,000
vₚ = 420.7 m/s
Step 2 · Find mean speed (⟨v⟩)
Use ⟨v⟩ = √(8 × 8.314 × T ÷ (π × M ÷ 1,000))
The
average
speed
over
the
whole
distribution,
√(8R
T
πM
);
the
fast
tail
lifts
it
above
the
peak.
⟨v⟩
=
√(8
×
8.314
×
298.15
π × 28.01 ÷ 1,000
)
⟨v⟩
=
√(8
×
8.314
×
298.15
0.088
)
⟨v⟩
=
√(66.512
×
298.15
0.088
)
⟨v⟩
=
√(
19,830.5528
0.088
)
⟨v⟩
=
√
225
,300
⟨v⟩ = 474.7 m/s
Step 3 · Find root-mean-square speed (vᵣₘₛ)
Use vᵣₘₛ = √(3 × 8.314 × T ÷ (M ÷ 1,000))
The
kinetic
energy
per
mole
is
(
3
2
)RT
=
(
1
2
)
M
⟨
v
2
⟩,
so
the
mean
squared
speed
is
3R
T
M
.
vᵣₘₛ
=
√(3
×
8.314
×
298.15
28.01 ÷ 1,000
)
vᵣₘₛ
=
√(3
×
8.314
×
298.15
0.02801
)
vᵣₘₛ
=
√(24.942
×
298.15
0.02801
)
vᵣₘₛ
=
√(
7,436.4573
0.02801
)
vᵣₘₛ
=
√
265
,500
vᵣₘₛ = 515.3 m/s
Check
Put the numbers back in. Both sides should match.
420.7
=
√(2
×
8.314
×
298.15
28.01 ÷ 1,000
)
✓
474.7
=
√(8
×
8.314
×
298.15
π × 28.01 ÷ 1,000
)
✓
515.3
=
√(3
×
8.314
×
298.15
28.01 ÷ 1,000
)
✓
Topic check
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