3.3. The big picture[[Student version, December 8, 2002]] 93
(the X-ray scattering work of Rosalind Franklin and Maurice Wilkins) with biochemical facts (the
base-composition rules observed by Erwin Chargaff), they deduced their now-famous double helix
model for the structure of DNA in 1953. The molecular biology revolution then began in earnest.
The big picture
Returning to the Focus Question, this chapter has explored the idea that random thermal motion
dominates the molecular world. We found that this idea explains quantitatively some of the behavior
of low-density gases. Gas theory may seem remote from the living systems we wish to study, but
in fact it turned out to be a good playing-field to develop some themes that transcend this setting.
Thus
- Section 3.1 developed many concepts from probability that will be needed later.
- Sections 3.2.3– 3.2.5 motivated three crucial ideas using ideal gases, namely the Boltz-
mann distribution, the Arrhenius rate law, and the origin of friction, all of which
will turn out to be general. - Section 3.3 also showed how the concept of activation barrier, on which the Arrhenius
law rests, led to the correct hypothesis that a long-chain molecule was the carrier of
genetic information.
Chapters 7 and 8 will develop the general concept of entropic forces, again starting with ideas from
gas theory. Even when we cannot neglect the interactions between particles, for example when
studying electrostatic interactions in solution, Chapter 7 will show that sometimes the noninteract-
ing framework of ideal-gas theory can still be used.
Key formulas
1.Probability: The mean value of any quantityfis〈f〉=∫
dxf(x)P(x)(Equation 3.9). The
variance is the mean-square deviation, variance(f)=〈f−〈f〉)^2 〉.
Addition rule: The probability to get either of two mutually exclusive outcomes is the sum
of the individual probabilities.
Multiplication rule: The probability to get particular outcomes in each of two independent
random strings is the product of the individual probabilities (Equation 3.15).
The Gaussian distribution isP(x)=√ 21 πσe−(x−x^0 )^2 /^2 σ^2 (Equation 3.8). The root-mean-
square deviation, ∆x,ofthis distribution equalsσ.
2.Thermal energy: The average kinetic energy of an ideal gas molecule at temperatureTis
3
2 kBT(Equation 3.21).
3.Maxwell: In a free, ideal gas the probability distribution for a molecule to havex-component
of velocity betweenvxand vx+dvx is a constant times e−m(vx)(^2) / 2 kBT
dvx. The total
distribution for all three components is then the product, namely another constant times
e−mv
(^2) / 2 kBT
d^3 v. The Maxwell distribution generalizes this statement for the case of many
particles (Equation 3.25).
4.Boltzmann: In an ideal gas on which forces act, the probability for one molecule to have
given position and momentum is a constant times e−E/kBTd^3 vd^3 x,where the total energy
Eof the molecule (kinetic plus potential) depends on position and velocity. In the special