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UNITS

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.m².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 ¹²₆C. 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 10²³, which is Avagadro's constant, N.

Electrical charge: A mole of singly-charged species has a charge of 9.648 x 10⁴ C (the faraday, F); 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 9.647 x 10⁴ x 0.1 = -19.29 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 10³       = 0.156 V
                    zF        2 x 9.647 x 10⁴

Under conditions where DG' for ATP hydrolysis is say -60 kJ.mol^-1, then DG^o'_elec = 0.312 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 9.647x 10⁴ x 0.24 J.mol^-1
= 69.46 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 10⁸ m.s^-1 (the velocity of light) and l is in meters.

If l is in nm, then

E = h x 2.998 x 10⁸ = 1.987 x 10^-16     J per photon
          l_nm x 10^-9           l_nm

A mole of photons is called an einstien, 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 10²³
                l_nm
      = 1.1962 x 10⁸ J.mol^-1
                l_nm

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

E   =     1.1962 x 10⁸               J.mol^-1/J.mol^-1.V^-1
          l_nm x 9.649 x 10⁴

  =   1240     eV
        l_nm

Thus, for photons of l = 500 nm,

E = 2.3924 x 10⁵ J mol^-1
    = 239.24 kJ.mol^-1
or

E = 2.48 eV.