Second Law of Thermodynamics - Entropy

The Second Law of Thermodynamics provides fundamental insights into the direction of spontaneous processes. It introduces the concept of entropy ($S$), a thermodynamic quantity that reflects the level of disorder or randomness in a system.

The second law states that in any isolated system, spontaneous processes tend to increase the entropy of the system. For any spontaneous process in an isolated system:

\[ \Delta S > 0 \]

This means that systems naturally evolve toward equilibrium, where entropy is maximized. Consider the example of burning wood in the presence of oxygen ($O_2$):

  • Spontaneous process: $C(s) + O_2(g) \rightarrow CO_2(g) + H_2O(l)$
  • Once initiated (by overcoming the activation barrier), this process releases energy and increases the entropy of the system.

 

Derivation

From the First Law of Thermodynamics, for a reversible process:

\[ \Delta Q_{rev} = dU - dW_{rev} \]

For an ideal gas, this equation can be expressed as:

\[ \Delta Q_{rev} = C_v(T) dT + p dV \]

Using the ideal gas law ($p = \frac{nRT}{V}$), we can substitute for pressure:

\[ \Delta Q_{rev} = C_v(T) dT + \frac{nRT}{V} dV \]

Dividing both sides of the equation by temperature $T$ gives the change in entropy for a reversible process:

\[ \frac{\Delta Q_{rev}}{T} = \frac{C_v(T) dT}{T} + \frac{nR dV}{V} \]

Since entropy is a state function (meaning its value depends only on the initial and final states of the system, not the path taken to get there), the total entropy change for a reversible process is given as a function of temperature ($T$) and volume ($V$) by:

\[ \Delta S(V, T) = \int_{\tau_i}^{\tau_f} \frac{C_v(T)}{T} dT + \int_{V_i}^{V_f} \frac{nR}{V} dV \]

This equation represents the total entropy change during a process in which both temperature and volume vary. The first term accounts for the temperature dependence, and the second term accounts for the change in volume.

If the heat capacity at constant volume, $C_v$, is independent of temperature, the entropy change simplifies to:

\[ \Delta S(V, T) = C_v \ln \frac{T_f}{T_i} + nR \ln \frac{V_f}{V_i} \]

 

Spontaneity and Entropy Change

The Second Law provides a criterion for spontaneity based on the change in entropy ($\Delta S$). For example, it explains why heat flows spontaneously from hot objects to cold ones.  If the entropy of the system increases, the process is spontaneous:

\[ \Delta S > 0 \quad \text{(spontaneous process)} \]

It is never observed for heat to spontaneously flow from a colder object to a hotter one, as this would decrease the system's entropy.

If the entropy change is zero, the system is in equilibrium:

\[ \Delta S = 0 \quad \text{(equilibrium)} \]

 

Let's now provide a mathematical background for the prediction of the direction of spontaneous processes.

Spontaneity and the Sign of $ \Delta S $

Consider heat flowing between two objects at different temperatures, $T_1$ and $T_2$ (with $T_1 > T_2$). When heat flows from the hotter object to the cooler one, the process is spontaneous, and the total entropy increases.

For an isolated system with heat flowing from a hotter region ($T_1$) to a colder region ($T_2$), we can express the heat flow as:

\[ \Delta Q_p = \Delta H \]

Since this is a reversible process, we can represent the heat flow as:

\[ \Delta Q_p = \Delta Q_{rev} \]

The heat transferred from the hotter body to the cooler body is:

\[ dQ_1 = -dQ_2 = dQ_p \]

The change in entropy for each part of the system is given by:

\[ dS = \frac{dQ_{rev,1}}{T_1} + \frac{dQ_{rev,2}}{T_2} \]

Substituting $dQ_1 = -dQ_2 = dQ_p$, we get:

\[ dS = dQ_p \left( \frac{1}{T_1} - \frac{1}{T_2} \right) \]

Since $T_1 > T_2$, the term $ \left( \frac{1}{T_1} - \frac{1}{T_2} \right) $ is positive, so $dS > 0$. This means that heat flows spontaneously from the hotter object to the cooler one, and the total entropy increases.

This way we arrive at the discussed conditions for spontaneous and non-spontaneous processes:

  • If $ \Delta Q_p < 0 $ (heat flows from hotter to colder), then $ \Delta S > 0 $, meaning the process is spontaneous.
  • If $ \Delta Q_p > 0 $ (heat flows from colder to hotter), then $ \Delta S < 0 $, meaning the process is non-spontaneous.

Therefore, for any process that occurs in an isolated system, there is a unique direction of spontaneous change defined by $ \Delta S > 0 $. In other words, entropy tends to increase in isolated systems, driving spontaneous processes.

 

Entropy Changes in Specific Processes

  • For a reversible process at constant pressure, the entropy change is:

\[ \Delta S = \frac{\Delta Q_p}{T} \]

  • Entropy is a state function, thus the result for reversible process is valid also for an irreversible. It follows also that the entropy change for a cyclic process is zero:

\[\Delta S = \oint \frac{dQ_{rev}}{T} = 0\]

  • For an isothermal process, the change in entropy is given by:

\[ \Delta S = nR \ln \frac{V_f}{V_i} \]

  • For an isochoric process, the change in entropy is given by:

\[\Delta S = \int \frac{dQ_{rev}}{T} = \int \frac{n C_{v,m} \, dT}{T} = n C_{v,m} \cdot \ln \frac{T_f}{T_i}\]

  • For an isobaric process, the change in entropy is given by:

\[\Delta S = \int \frac{\Delta Q_{rev}}{T} = \int \frac{n C_{p,m} \, dT}{T} = n C_{p,m} \ln \frac{T_f}{T_i}\]

  • For an adiabatic process, no heat is transferred, and thus the entropy change is zero:

\[ \Delta S = 0 \quad \text{(adiabatic process)} \]

 

Standard Entropy

Standard entropy, denoted as $S^\circ$, is the absolute entropy of a substance measured at standard conditions. It is especially useful for chemical reactions, as it allows for the calculation of the standard entropy change for a reaction, $\Delta S^\circ_{\text{reaction}}$:

\[ \Delta S^\circ_{\text{reaction}} = \sum S^\circ_{\text{products}} - \sum S^\circ_{\text{reactants}} \]

 

Standard Entropy of a Reaction and its Temperature Dependence:

\[aA + bB \rightarrow cC + dD\]

\[\Delta_{rxn} S^\circ = \sum \nu S_m^\circ (products) - \sum \nu S_m^\circ (reactants)\]

where $\nu_j$ represents stoichiometric coefficients (positive for products, negative for reactants).

Temperature Dependence of Entropy (derrived from Kirchhoff's Law):

\[\Delta_{rxn} S^\circ (T_2) = \Delta_{rxn} S^\circ (T_1) + \int_{T_1}^{T_2} \frac{c_{p,m}}{T} \, dT\]

 

Entropy in Phase Transitions:

When a substance undergoes a phase transition (e.g., solid to liquid or liquid to gas), its entropy changes dramatically. For example, the transition from solid ice to liquid water involves a large increase in entropy because the liquid phase is more disordered than the solid phase.

Phase transitions occur through heat exchange at $T = const.$ and $ p = const. $:

\[\Delta_{vap} S = \int \frac{dQ_{rev}}{T} = \frac{\Delta Q_{rev}}{T_{vap}} = \frac{\Delta_{vap} H}{T_{vap}}\]

 

\[\Delta_{fus} S = \int \frac{dQ_{rev}}{T} = \frac{\Delta Q_{rev}}{T_{fus}} = \frac{\Delta_{fus} H}{T_{fus}}\]

 

Clausius Inequality

The discussion above provided a framework for establishing entropy from the perspective of reversible processes. However, the mojority of processes are irreversible and thus we need to provide a general formulation. The Clausius Inequality arises from the broader implications of the Second Law of Thermodynamics and serves as a cornerstone in understanding spontaneous processes, entropy, and the limits of thermodynamic systems. To develop an intuitive grasp of this inequality, let's begin by exploring the underlying principles of reversible and irreversible processes, how they relate to work and heat, and how these relationships culminate in the Clausius Inequality.

Work in Reversible and Irreversible Processes

In thermodynamics, a fundamental distinction exists between reversible and irreversible processes. A reversible process is idealized—it proceeds infinitely slowly and without friction, so the system remains in equilibrium throughout. In contrast, real processes are often irreversible, involving dissipative effects like friction or turbulence.

The system does more work in a reversible process than in an irreversible one. Mathematically, we can express this relationship as:

\[ |dW_{rev}| > |dW_{irr}| \]

This inequality means that for the same change in state, more work can be extracted from a reversible process than an irreversible one. The additional work extracted in reversible processes corresponds to a lower degree of entropy generation.

Internal Energy and Heat Exchange

The First Law of Thermodynamics holds for both reversible and irreversible processes. However, while the internal energy (state function) change $dU$ remains the same regardless of the path (reversible or irreversible), the heat exchanged, and the work done depend on the process type.:

\[ dU = dQ_{rev} - p_{ext} \, dV = dQ_{rev} - p \, dV \]

Simplifying this, we get the following relationship between heat for reversible and irreversible processes:

\[ dQ_{rev} - dQ_{irr} = (p - p_{ext}) dV \]

This difference explains how the pressure difference between the system ($p$) and the external environment ($p_{ext}$) affects the process:

  • If $p - p_{ext} > 0$, spontaneous expansion occurs ($dV > 0$).
  • If $p - p_{ext} < 0$, spontaneous compression occurs ($dV < 0$).

This leads to the general condition for spontaneous processes:

\[ (p - p_{ext}) dV > 0 \]

This condition indicates that spontaneous expansion or compression depends on the pressure difference and the change in volume.

Furthermore, since the system does more work in a reversible proces, and thus $ |dW_{rev}| > |dW_{irr}| $, then $dQ_{rev} > dQ_{irr}$. Therefore:

\[ \frac{Q_{rev}}{T} \geq \frac{Q_{irr}}{T} \implies \Delta S \geq \frac{Q_{rev}}{T} \]

The Clausius Inequality provides a quantitative way to describe the Second Law of Thermodynamics. It states that more work is done when a change is reversible than when it is irreversible. For any system, the inequality is expressed as:

\[ dS \geq \frac{dQ}{T} \]

This leads to the conclusion that spontaneous processes are generally irreversible.

Example: cooling is a spontaneous process:

When heat flows from a hot object to a cold object, it is a spontaneous process, which results in an increase in the total entropy of the system. The following relationships describe the process:

\[ dS \geq \frac{dQ_h}{T_h} \quad \text{and} \quad dS \geq \frac{dQ_c}{T_c} \]

Therefore, the total change in entropy is given by:

\[ dS \geq \frac{dQ_h}{T_h} + \frac{dQ_c}{T_c} \]

Since the heat lost by the hot object equals the heat gained by the cold object ($dQ_h = - dQ_c$), we can rewrite the equation as:

\[ dS \geq - \frac{dQ_c}{T_h} + \frac{dQ_c}{T_c} \]

Factoring out $dQ_c$, we get:

\[ dS \geq \left( \frac{1}{T_c} - \frac{1}{T_h} \right) dQ_c \]

Since $dQ_c > 0$ and $T_h \geq T_c$, we conclude that:

\[ dS \geq 0 \]

This confirms that the process is spontaneous because the total entropy change is positive.

The Clausius Inequality has profound implications for various thermodynamic processes, especially when applied to engines, refrigerators, and other cyclic systems. It sets the upper limit for efficiency in heat engines and provides a criterion for determining the direction of natural processes. In practical terms, it means that real systems, which are always irreversible to some extent, will generate entropy and thus cannot operate at the ideal efficiency of a reversible engine. Therefore, it provides a powerful tool for analyzing the limits of energy conversion processes in real-world systems.