SME
Biochemistry · Problem type
Free energy of the proton gradient
Use this for “A mitochondrion has Δψ = 0.170 V, the matrix negative, and its matrix pH is 0.50 above the outside. Find ΔG for each H⁺ that flows back in at 37 °C.”
Diagram
V = −150 mV: the matrix is negative, so + charges line the intermembrane space’s face and − the matrix’s.
One pair per 10 mV: 8 a side (8 at most).
Dots on one scale, one dot ≈ 0.025, relative to the intermembrane space (= 1): H⁺ 1 in the intermembrane space, 0.178 in the matrix.
This is an example. Type your own values to start a new problem.
ΔG (kJ/mol)
=
−F
×
Δψ (V)
−
2.303R
T (K)
×
ΔpH
Gₑₗ
Electrical part
calculated · kJ/mol
GpH
Chemical part
calculated · kJ/mol
Formulas
Gₑₗ = 96.485 × Δψ
+14.47 = 96.485 × 0.15
GpH
=
2.303
×
8.314
×
T
×
ΔpH
1,000
+4.454
=
2.303
×
8.314
×
310.15
×
0.75
1,000
ΔG = −(Gₑₗ + GpH)
−18.93 = −(14.47 + 4.454)
Assumptions
•
Δψ is outside minus inside (positive when the matrix is negative); ΔpH is pH inside minus pH outside.
•
A mole of H⁺ moving in gains FΔψ from the charge and 2.303RTΔpH from the concentration, both in kJ/mol.
•
ΔG = −(FΔψ + 2.303RTΔpH) for H⁺ flowing in; pumping it out costs the same amount.
•
F = 96.485 kJ/(mol·V) and R = 8.314 J/(mol·K) (÷ 1000 for kJ); body temperature, 37 °C, is 310.15 K.
Step-by-step
WE KNOW
Δψ = +0.15 V, ΔpH = +0.75, T = 310.15 K
FIND
electrical part (Gₑₗ), chemical part (GpH), free energy per H⁺ moved in (ΔG)
Step 1 · Find electrical part (Gₑₗ)
Use Gₑₗ = 96.485 × Δψ
A mole of H⁺ carries F = 96.485 kJ for each volt it crosses: multiply F by Δψ.
Gₑₗ = 96.485 × 0.15
Gₑₗ = +14.47 kJ/mol
Step 2 · Find chemical part (GpH)
Use GpH = 2.303 × 8.314 × T × ΔpH ÷ 1,000
Each
pH
unit
is
a
tenfold
step
in
[H⁺],
worth
2.303RT;
multiply
by
ΔpH
and
1000
for
kJ/mol.
GpH
=
2.303
×
8.314
×
310.15
×
0.75
1,000
GpH
=
19.147
×
310.15
×
0.75
1,000
GpH
=
5,938.4
×
0.75
1,000
GpH
=
4,453.8
1,000
GpH = +4.454 kJ/mol
Step 3 · Find free energy per H⁺ moved in (ΔG)
Use ΔG = −(Gₑₗ + GpH)
Both parts push H⁺ inward, so its free energy drops by their sum as it flows in.
ΔG = −(14.473 + 4.4539)
ΔG = −18.93 kJ/mol → ΔG < 0: H⁺ runs back in on its own, and ATP synthase turns that drop into ATP.
Check
Put the numbers back in. Both sides should match.
+14.4728 = 96.485 × 0.15 ✓
+4.4539
=
2.303
×
8.314
×
310.15
×
0.75
1,000
✓
−18.93 = −(14.473 + 4.4539) ✓
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