[Iamyourfather]

Electron transfer translocates protons from the mitochondrial matrix to the external medium, establishing a pH gradient across the inner membrane (outside more acidic than inside). The tendency of protons to diffuse back into the matrix is the driving force for ATP synthesis by ATP synthase. During oxidative phosphorylation by a suspension of mitochondria in a medium of pH 7.4, the pH of the matrix has been measured as 7.7.

This is what I've done so far (correct me if I've made any errors):

a)   Calculate [H+] in the external medium and in the matrix under these conditions (10%)

-log10 [H+] = PH; inverse = [H+] =10-PH

[H+] in external medium = 10-7.4 = 4.0 x 10-8 M

[H+] in matrix = 10-7.7 = 2.0 x 10-8 M


b)   What is the outside-to-inside ratio of [H+]? Comment on the energy inherent in this concentration difference.

The outside-to-inside ratio is 2:1. The free energy inherent in this concentration difference can be calculated across the inner membrane at a temperature of 25° C:

K = (C2 /C1) = 2 M0   
ΔG’o = -RT In K
ΔG'° = ΔG' = -RT In K/Γ
Γ = 1 M/1 M = 1 M0
ΔG’o = -RT In (C2 /C1) = -2476 In (2) = -1716 J mol-1 = -1.72 KJ mol-1

c)   From these data, is the pH gradient alone sufficient to generate ATP? Explain

The energy available from the H+ concentration gradient, 2.3ΔpH RT = 2.3(0.3) (2476) = 1.71 KJ mol-1

What is the free energy needed to generate ATP?

Is the answer from b) and c) meant to me the same? If so my are my answers different my 0.01?

[Babcock_Hall]

I would look up or calculate the free energy needed to synthesize ATP from ADP and phosphate and compare that value with your calculation in part (b).  BTW, the two values you found are the same within a rounding error, because you basically calculated the same thing in two slightly different ways.

[Iamyourfather]

I would look up or calculate the free energy needed to synthesize ATP from ADP and phosphate and compare that value with your calculation in part (b).  BTW, the two values you found are the same within a rounding error, because you basically calculated the same thing in two slightly different ways.

Isn't -2.3ΔpH RT more accurate than 2.3ΔpH RT?

[Babcock_Hall]

I did not see the sign difference in the two equations and the two answers until just now.  I believe that -2.303RT(ΔpH) is correct, with respect to the concentration difference.  However, this number is likely to much smaller than the free energy of hydrolysis of ATP (have you looked that up?).  And the complete equation for ΔG of transport across a membrane has a second term.