294 Chapter 8. Chemical forces and self-assembly[[Student version, January 17, 2003]]
T 2 Track 2
8.1.1′
- Equation 8.1 on page 260 definedμas a derivative with respect to the number of moleculesN.
Chemistry textbooks instead defineμas a derivative with respect to the “amount of substance”n.
See the discussion of units in Section 1.5.4′on page 26. - The discussion in the Example on page 261 amounted to converting a derivative taken with
Ekinfixed to one withEfixed. The formal way to summarize this manipulation is to say that
∂S
∂N
∣∣
E=
∂S
∂N∣∣
Ekin−∂S
∂Ekin∣∣
∣N.
- We have been describingas if it were a form of potential energy, like a coiled spring inside
the molecule. Purists will insist that the energy of a chemical bond is partly potential and partly
kinetic, by the Uncertainty Principle. It’s true. What lets us lump these energies together, indeed
what lets us speak of bond energies at all, is that quantum mechanics tells us that any molecule
at rest has aground statewith a fixed, definite total energy. Any additional kinetic energy from
center-of-mass motion, and any potential energy from external fields, are given by the usual classical
formulas and simply added to the fixed internal energy. That’s why we get to use familiar results
from classical physics in our analysis. - A complicated molecule may have many states of almost equally low energy. Thenwill have
atemperature-dependent component reflecting in part the likelihood of occupying the various low-
energy states. But we won’t usedirectly; we’ll useμ^0 ,which we already knew was temperature-
dependent anyway. This fine point doesn’t usually matter because living organisms operate at
nearly fixed temperature; once again our attitude is thatμ^0 is a phenomenological quantity.
8.2.1′ There are other, equivalent definitions ofμbesides the one given in Equation 8.1 on page 260.
Thus for example some advanced textbooks state your results from Your Turn 8c asμ=∂F∂N
∣∣
∂G T,V=
∂N∣∣
T,p.Twomore expressions for the chemical potential are∂E
∂N∣∣
S,V and∂H
∂N∣∣
S,p,whereHis the
enthalpy. The definition in Equation 8.1 was chosen as our starting point because it emphasizes
the key role of entropy in determining any reaction’s direction.
8.2.2′
- The solutions of interest in cell biology are frequentlynotdilute. In this case the Second Law still
determines a reaction’s equilibrium point, but we must use the activity in place of the concentration
[X] when writing the Mass Action rule (see Section 8.1.1 on page 260). Dilute-solution formulas
are epecially problematic in the case of ionic solutions (salts). That’s because our formulas ignore
the electrostatic interaction between ions (and indeed all other interactions). Since the electrostatic
interaction is of long range, its omission becomes a serious problem sooner than that of other
interactions. See the discussion in Landau & Lifshitz, 1980,§92. - We can also think of the temperature dependence of the equilibrium constant (Your Turn 8d on
page 266) as an instance of Le Chˆatelier. Dumping thermal energy into a closed system increases the
temperature (thermal energy becomes more available). This change shifts the equilibrium toward
the higher-energy side of the reaction. By absorbing thermal energy, the shift makes the actual
increase of the system’s temperature smaller than it would have been if no reactions were allowed,
partially undoing our original disturbance.