Sections#

Review & Summary#

One-Way Processes#

An irreversible process is one that cannot be reversed by means of small changes in the environment. The direction is set by the change in entropy \(\Delta S\) of the system. Entropy \(S\) is a state property; it depends only on the state of the system. The entropy postulate: If an irreversible process occurs in a closed system, the entropy of the system always increases.

Calculating Entropy Change#

The entropy change \(\Delta S\) for an irreversible process from state \(i\) to \(f\) equals the entropy change for any reversible process between those same states. We can compute the latter with

(188)#\[ \Delta S = S_f - S_i = \int_i^f \frac{dQ}{T} \]

where \(Q\) is the energy transferred as heat and \(T\) is the temperature in kelvins. For a reversible isothermal process, \(\Delta S = Q/T\). When an ideal gas changes reversibly from \((T_i, V_i)\) to \((T_f, V_f)\),

(189)#\[ \Delta S = n C_V \ln\frac{T_f}{T_i} + n R \ln\frac{V_f}{V_i} \]

The Second Law of Thermodynamics#

The entropy of a closed system increases for irreversible processes and remains constant for reversible processes. It never decreases: \(\Delta S \geq 0\).

Engines#

An engine is a device that, operating in a cycle, extracts energy as heat \(|Q_H|\) from a high-temperature reservoir and does work \(|W|\). The efficiency is

(190)#\[ \varepsilon = \frac{|W|}{|Q_H|} = 1 - \frac{|Q_C|}{|Q_H|} \]

where \(|Q_C|\) is the energy discharged as heat to the low-temperature reservoir. A Carnot engine is an ideal reversible engine. Its efficiency is

(191)#\[ \varepsilon_C = 1 - \frac{T_L}{T_H} \]

where \(T_H\) and \(T_L\) are the reservoir temperatures. A perfect engine (converting heat entirely to work) would violate the second law.

Refrigerators#

A refrigerator extracts energy \(|Q_L|\) as heat from a low-temperature reservoir with work \(W\) done on it. The coefficient of performance is \(K = |Q_L|/W\). For a Carnot refrigerator, \(K_C = T_L/(T_H - T_L)\). A perfect refrigerator (transferring heat from cold to hot without work) would violate the second law.

Entropy from a Statistical View#

The entropy can be defined in terms of microstates (possible distributions of molecules). The multiplicity \(W\) is the number of microstates in a configuration. For a system of \(N\) molecules,

(192)#\[ S = k \ln W \]

where \(k\) is the Boltzmann constant. This connects thermodynamics to the microscopic behavior of molecules.