Abstract
This chapter develops a microscopic, statistical view of temperature, heat, work, and the laws of thermodynamics. A key concept is the Boltzmann factor, relating the thermal population of microstates to their energy, and its related concept the Nernst equation. The principle of equipartition of energy leads us to an analysis of the Brownian motion of small particles immersed in a fluid. When two systems can exchange particles as well as energy, the chemical potential governs their behavior. The Gibbs free energy is used to describe chemical reactions that take place at constant temperature and pressure. It is closely related to the chemical potential.
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Notes
- 1.
In computational biology, a mole of differential equations is sometimes called a leibniz (Huang and Wikswo 2004). Solving for the motion of each water molecule in a cubic millimeter of blood requires solving 0.16 millileibniz of equations.
- 2.
- 3.
A good book on probability is Weaver (1991).
- 4.
A system is that part of the universe that we choose to examine. The surroundings are the rest of the universe. The system may or may not be isolated from the surroundings.
- 5.
There is a subtlety about the meaning of average that we are glossing over here. If we take a whole ensemble of identical systems, which were all prepared the same way, and measure \(n\) in each one, we have the ensemble average \(\bar {n}\). This is calculated in the way described in Appendix G. If we watch one system over some long time interval, as in Fig. 3.4, we can take the time average \(\left \langle n\right \rangle \). It is taken by recording values of \(n\) for a large number of discrete times in some interval. Strictly speaking, an equilibrium state is one in which the ensemble average is not changing with time.
- 6.
A more detailed discussion of equilibrium states is found in Reif (1994).
- 7.
For a more detailed discussion of these assumptions, see Reif (1994, Chap. 3).
- 8.
In thermodynamics and statistical mechanics, equilibrium and steady state do not mean the same thing. Steady state means that some variable is not changing with time. The concentration of sodium in a salt solution flowing through a pipe could be in steady state as the solution flowed through, but the system would not be in equilibrium. Only a few microstates corresponding to bulk motion of the fluid are occupied. In other areas, such as feedback systems, the words equilibrium and steady state are used almost interchangeably.
- 9.
If \(\Omega \) is a continuous function of \(U\), then \(\Omega (U)dU\) is actually the number of states wih energy between \(U\) and \(U+dU\). We ignore this distinction. For a discussion of it, see Chap. 3 of Reif (1994).
- 10.
A more detailed justification of this is found in earlier editions of this book, in texts on statistical mechanics, or on the web site associated with this book.
- 11.
In this book, \(N\) represents the number of particles, and the chemical potential has units of energy per particle. In other books it may have units of energy per mole.
- 12.
An ideal solution is defined in Sect. 3.18.
- 13.
There are multiple pathways in glucose respiration. The 36 is approximate.
- 14.
- 15.
- 16.
The fact that there is only one microstate because of the indistinguishability of the particles is called the Gibbs paradox (Reif 1998).
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Hobbie, R., Roth, B. (2015). Systems of Many Particles. In: Intermediate Physics for Medicine and Biology. Springer, Cham. https://doi.org/10.1007/978-3-319-12682-1_3
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