SME
Physical Chemistry II: Quantum & Spectroscopy
Magnetic resonance (NMR and EPR)
Use this for “¹³C has γ = 6.728 × 10⁷ rad s⁻¹ T⁻¹. Find its resonance frequency in a 14.10 T magnet and ΔE ÷ kT at 298 K.”
Chart
ν(9.4) ≈ 400.2
This is an example. Type your own values to start a new problem.
ν (MHz)
=
γ (rad/(s·T))
×
B₀ (T)
÷
2π
ΔE
Energy gap
calculated · J
T
Temperature
example · K
ΔE ÷ kT
Energy gap over thermal energy
calculated
Formulas
ν
=
γ
×
B₀
2π ×
10
6
400.2
=
2.675 ×
10
8
×
9.4
2π ×
10
6
ΔE
=
6.626 ×
10
-34
×
ν
×
10
6
2.6517 ×
10
-25
=
6.626 ×
10
-34
×
400.2 ×
10
6
ΔE
kT
=
ΔE
1.381 ×
10
-23
×
T
6.4401 ×
10
-5
=
2.6517 ×
10
-25
1.381 ×
10
-23
× 298.15
Assumptions
•
A spin-½ nucleus in a field B₀ has two levels; radio waves at ν = γB₀ ÷ 2π flip the spin from the lower level to the upper one.
•
γ is taken as its size, in rad s⁻¹ T⁻¹; ¹H has 2.675 × 10⁸ and ¹³C has 6.728 × 10⁷.
•
ΔE ÷ kT is tiny, so the lower level holds more than half the nuclei by only about ΔE ÷ 2kT of them.
•
h = 6.626 × 10⁻³⁴ J·s and kB = 1.381 × 10⁻²³ J/K.
Step-by-step
WE KNOW
γ = 2.675 × 10⁸ rad/(s·T), B₀ = 9.4 T, T = 298.15 K
FIND
resonance frequency (ν), energy gap (ΔE), energy gap over thermal energy (ΔE ÷ kT)
Step 1 · Find resonance frequency (ν)
Use ν = γ × B₀ ÷ (2π × 10⁶)
γB₀
is
the
angular
frequency
in
rad/s;
divide
by
2π
for
hertz
and
by
10
6
for
megahertz.
ν
=
2.675 ×
10
8
×
9.4
2π ×
10
6
ν
=
400.2
MHz
Step 2 · Find energy gap (ΔE)
Use ΔE = 6.626 × 10⁻³⁴ × ν × 10⁶
The photon that flips the spin carries the gap: Planck’s constant times the frequency in hertz.
ΔE
=
6.626 ×
10
-34
×
400.2 ×
10
6
ΔE
=
2.6517 ×
10
-25
J
Step 3 · Find energy gap over thermal energy (ΔE ÷ kT)
Use ΔE ÷ kT = ΔE ÷ (1.381 × 10⁻²³ × T)
Compare the gap with the thermal energy kT.
ΔE
kT
=
2.6517 ×
10
-25
1.381 ×
10
-23
× 298.15
ΔE
kT
=
2.6517 ×
10
-25
4.1175 ×
10
-21
ΔE
kT
=
6.4401 ×
10
-5
→
Out
of
every
million
nuclei,
the
lower
level
holds
only
about
32
more
than
half:
the
NMR
signal
is
weak.
Check
Put the numbers back in. Both sides should match.
400.2
=
2.675 ×
10
8
×
9.4
2π ×
10
6
✓
2.6517 ×
10
-25
=
6.626 ×
10
-34
×
400.2 ×
10
6
✓
6.4401 ×
10
-5
=
2.6517 ×
10
-25
1.381 ×
10
-23
× 298.15
✓
Topic check
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