Mechanically, a joule is equivalent to the energy dissipated by a force of one newton acting through 1 meter. Work is given by (w = force x distance); since force = mass x acceleration, with base units kg.m.s^{-2} (defining the derived unit of a newton (N)), 1 joule = N.m with base units of kg.m^{2}.s^{-2}.

Electrically, a joule is equivalent to the energy dissipated by a current of one ampere (A) passing though a potential of one volt (V) for one second. Since the derived unit of a volt is defined with units W.A^{-1}, and the derived unit of a watt (W) is defined in J.s^{-1}, we get overall A.J.s^{-1}.A^{-1}.s, giving J = A.V.s.

1 cal = 4.184 J.

**Substance**: the amount of a substance is measured by the mole (mol); the mole is defined as the amount of substance of a system which contains as many __entities__ as there are atoms in 0.012 kg of the carbon isotope ^{12}_{6}C. __Entities__ may be atoms, molecules, ions, electrons, or other particles or groups of particles.

The number of entities in a mole is 6.022 x 10^{23}, which is __Avagadro's constant__, *N*.

**Electrical charge**: A mole of singly-charged species has a charge of 9.64853415 x 10^{4 }C (the faraday, F, rounded to 9.649 x 10^{4 }C in this course); a Coulomb (C) is the charge carried by a current of one of the ampere in one second. So C has base units of amp.s and F has base units of amp.s .mol^{-1}.

One singly charged species (an electron or H^{+}, for example) has a charge of 1.602 x 10^{-19 } C, so this multiplied by Avagadro's constant is the faraday (F).

**Gas Constant**: R has the value 8.314 J.K^{-1}.mol^{-1} (or 1.987 cal.K^{-1}.mol^{-1}).

So RT has the units J.mol^{-1}

The units for RT/F are obtained as follows:

__RT__ = __J.mol__^{-1} = __amp.volt.s.mol__^{-1} = volts

F amp.s.mol^{-1} amp.s.mol^{-1}

**Redox span and DG**

The Farady constant provides a factor relating electrical potential difference to free energy. For two redox couples differing in potential by an amount DE', the free energy per mol which is available from electron transfer between them is given by:

DG' = -zF.DE' (J.mol^{-1})

DE' = __-DG'__ (volts)

zF

where z is the number of redox equivalents involved in the reaction (usually z = 2 (for quinones, NAD+/NADH, etc.), or z = 1 (for most cytochromes)), and DE' means (E'_{(oxidizing couple)} - E'_{(reducing couple)}). Here the convention determines that the redox change is spontaneous (DG' is negative) for a reaction in which the oxidizing couple has a higher E' than the reducing couple.
Thus, if DE' = 0.1 volt, then for a reaction involving 2 equivalents:

DG' = -2 x 9.649 x 10^{4} x 0.1
= -19.298 kJ.mol^{-1}

It is permissible and sometimes useful to express the energy of change of other processes in electrical units. Thus the standard free energy of hydrolysis (DG^{o}') of ATP is approximately -30 kJ.mol^{-1}. When considering the proton gradient that can be driven by this work term, where Dp is expressed in V, it is useful to express this in electrical units. If this energy was used to drive a process involving 2 charges (e.g., 2 e^{-} or 2 H^{+}):

DG^{o}'_{elec} = -__DG__^{o}' = __ 30 x 10__^{3}__ __ = 0.156 V

zF 2 x 9.649 x 10^{4}

Under conditions where DG' for ATP hydrolysis is say -60 kJ.mol^{-1}, then
DG^{o}'_{elec} = 0.311 volts.

**Dp and DG' for ATP hydrolysis.**

If a proton motive force (Dp) is in equilibrium with a phosphorylation system in which the free energy for ATP synthesis is DG', then:

DG' = aF.Dp (J.mol^{-1})

where a is the overall stoichiometry of the proton pump. Taking a=3 H^{+}/ATP and Dp=0.24 volt, a typical value, then:

DG' = 3 x 9.649x 10^{4} x 0.24 J.mol^{-1}

= 69.47 kJ.mol^{-1
}

**Light**

Energy of a photon is given by Plank's equation:

E = hn (J)

where h = 6.626 x 10^{-34} J.s; n is the frequency (s^{-1}). Most usually in photosynthetic work, l (wavelength) is used in place of n. Then Plank's equation becomes:

E = hn = h c/l

where c = 2.998 x 10^{8} m.s^{-1} (the velocity of light) and l is in meters.

If l is in nm, then

E = __ h x 2.998 x 10__^{8} = __1.987 x 10__^{-16} J per photon

l_{nm} x 10^{-9}
l_{nm}

A mole of photons is called an einstein, and its energy is calculated by multiplying the above value by Avagadro's constant:

Energy per einstein = E x N

= __1.987 x 10__^{-16} x 6.022 x 10^{23}

l_{nm}

= __1.1962 x 10__^{8} J.mol^{-1}

l_{nm}

or expressing this in electron volts by dividing by the Faraday constant:

E = __ 1.1962 x 10__^{8}__ __ J.mol^{-1}/J.mol^{-1}.V^{-1}

l_{nm} x 9.649 x 10^{4}

= __ 1240 __ eV

l_{nm}

Thus, for photons of l = 500 nm,

E = 2.3924 x 10^{5} J mol^{-1}

= 239.24 kJ.mol^{-1}

or

E = 2.48 eV.

©Copyright 1996, Antony Crofts, University of Illinois at Urbana-Champaign, a-crofts@uiuc.edu