Chapter 4

Chapter 4

The statistical

interpretation of entropy

4.1 Introduction

• From the formal statement of the Second Law of Thermodynamics, it is difficult to assign a physical

significance or a physical quality to entropy. (different from internal energy, U)

• The 2

law- law only in that it has not been disproved.

Evidence? No perpetual motion machine.

• The physical interpretation of entropy had to await the

development of quantum theory and statistical mechanics

4.2 Entropy and disorder on an atomic scale

• The more mix-up the constituent particles in the system, the larger the value of its entropy. (Gibbs)

• Atomic arrangement from solid, to liquid, to gas: more disorder (entropy increases)

• Thus, S, can be related with atomic scale randomness or disorder

• If the melting process (more randomness of atoms) is conducted at constant pressure

∆S > 0 more mix-up-ness

4.2 Entropy and disorder on an atomic scale

• In super-cooled liquid case: spontaneous freeze (irreversible), it appears decrease in the degree disorder (l → s).

• Entropy has to increase due to ∆S

> 0

• Thus, it is not true if we consider the degree of order of the heat bath:

Increase in ordering is less tan increase in disordering of bath: ∆S

> 0

4.2 Entropy and disorder on an atomic scale

• If the freezing occurs at the equilibrium melting point,

then, the increase in the degree of order of the freezing system = the decrease in the degree of order of the heat reservoir absorbing the heat of solidification

• ∆s

= 0

• The equilibrium melting or freezing temperature = temp. at which no

change in the degree of order of the system + heat bath occurs. Only at

this temperature the solid phase is in equilibrium with the liquid phase.

4.3 The concept of microstate

• The effect of the quantization of energy with a perfect crystal: The crystal contains n

particles and has he energy, U. Then statistical mechanics asks this: (1) in how many ways can the n particles be distributed over the available energy levels such that the total energy of the crystal is U, and (2) which is the most probable?

• Consider that the crystal contains 3 identical particles located on 3 indistinguishable sites A, B, and C. If the energy levels are ε₀ (=0), ε₁ (=u), ε₂ (=2u) and ε₃ (=3u) and total energy is U=3u (4 energy levels available), the answer will be..

4.3 The concept of microstate

Total 10 microstates correspond to one macrostate!

(1) Representative cases to meet the U requirement (2) Possible # of combination for each case

4.4 Determination of the most probable microstate

• The number of arrangement within a given distribution, Ω, can be calculated.

• If n particles are distributed among the energy level such that n₀ in level ε₀, n₁ in level ε₁, n_r in level ε_r ….

(eq 4.1)

(eq 4.2)

4.4 Determination of the most probable microstate

(eq 4.3) As the macrostate of a system is determined by the fixed values of U, V and n, any distribution of particles among the energy levels must conform with the following conditions.

U = constant = n₀ε₀ + n₁ε₁ + n₂ε₂ ……..+ n_rε_r ….

n = constant = n₀ + n₁ + n₂ +……..+ n_r …. (eq 4.4)

Thus, δU = Σ δn_iε₁= 0 and δn = Σ δn_i = 0

(eq 4.5) (eq 4.6)

4.4 Determination of the most probable microstate

(eq 4.8) (eq 4.7)

4.4 Determination of the most probable microstate

(eq 4.5)

(eq 4.8) (eq 4.6)

(eq 4.10) (eq 4.9)

4.4 Determination of the most probable microstate

(eq 4.13) (eq 4.12) (eq 4.11)

4.4 Determination of the most probable microstate

The distribution of particles in the energy levels for max. Ω (the most probable

distribution) is the one which the occupancy of the levels decreases exponentially with ε_i and actually be determined by β.

β = 1/kT (eq 4.14)

4.5 The influence of temperature

(eq 4.15) As T increases, β decreases, where β = 1/kT. Upper levels become

more populated since avg. energy of the particles increases (increase in U/n). For fixed V, n: U increases as T does.

• When the number of particles is extremely large, total number of arrangements can be equated with max number of arrangements. i.e., ln Ω_max ~ ln Ω _total

P ?

4.5 The influence of temperature

(eq 4.15)

4.6 Thermal equilibrium and the Boltzman equation

When the exchange of energy carried out at constant total volume (eq 4.16)

Consider a system of particles in thermal equilibrium with a heat bath and let the sate of the combined system be fixed by fixing U, V, and n where,

4.6 Thermal equilibrium and the Boltzman equation

When the exchange of heat occurs at constant temperature and occurs reversibly

As both entropy and number of arrangements are state functions (eq 4.16)

(eq 4.17)

4.7 Heat flow and the production of entropy

• The condition for A to be in thermal equilibrium with B is

• And if total energy remains constant

(eq 4.18)

(eq 4.19)

(eq 4.20)

4.7 Heat flow and the production of entropy

• When a quantity of heat is transferred from A to B at total constant energy

• So if the temperature of A and B is same, the equation will be zero.

(eq 4.21)

4.8 Configurational entropy and thermal entropy

4.8 Configurational entropy and thermal entropy

4.8 Configurational entropy and thermal entropy

• If 𝑛𝑛

atoms of A are mixed with 𝑛𝑛

atoms of B

(eq 4.22)

(eq 4.23)

4.8 Configurational entropy and thermal entropy

• The total entropy of a system consists of its thermal entropy(𝑆𝑆_𝑡𝑡𝑡) and its configurational entropy(𝑆𝑆_{𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐})

• The number of spatial configurations available to two closed systems placed in thermal contact is unity.

(eq 4.24)