488
10 The Statistical Basis of Bioenergetics
• Small active proteins for pharmaceutical medicines;
• Injectable proteins to replace or supplement natural proteins, such as hormones;
• Injectable designed antibodies, such as against cancer cells;
• Synthetic membranes, both passive and active, for selective diffusion, such as for
water purification;
• Proteins that aid medical diagnosis, such as ones that fluoresce on specific
attachment;
• Proteins that react to the presence of very small quantities of a gas or particles in
air;
• Catalysts in the production of useful biochemical substances.
Fluctuations in the Particle Number
Systems in a Gibbs ensemble at or near equilibrium that are described by the
grand canonical ensemble have a fixed temperature and volume. But the number of
particles in any one system can vary, since particles can diffuse through permeable
walls between the systems. We now can ask what fluctuations in particle numbers
can we expect in one of these systems when it is at or near equilibrium? To answer,
we will calculate the root-mean-square of the N , called N. We have
((N)
2
=
(N − N )
2
=
N
2
− N
2
(10.181)
=
1
∂ 2
∂γ 2 −
1
∂ ∂
∂γ
2
=
∂
∂γ
1
∂ ∂
∂γ
=
∂N
∂γ
T
.
(10.182)
To express the result in terms of more commonly measured thermodynamic
variables, we first notice that changes of N cause changes of the particle density,
ρ ≡ N/V . In these terms (and using μ = k B T γ ),
∂N
∂γ
T
= k B T V
∂ρ
∂μ
T
.
The Gibbs–Duhem relation Eq. (9.82) connects changes of μ with changes of
pressure and temperature:
Ndμ = V dp − S dT .
In the present case, the temperature is constant, so dμ = (1/ρ)dp, which makes