Entropy Changes in Thermodynamic Processes (Closed Systems)

Entropy Change of a Closed System

Entropy is a thermodynamic property (state function), meaning it has a definite value at each state of a system.

Therefore, the change in entropy between two specified states is independent of the process path.
For any general process, the entropy change of a system is expressed using the entropy balance:

Heat transfer and irreversibilities cause the change in entropy during a process.
For a reversible process, there are no irreversibilities; hence entropy generation term is zero (S_gen,sys = 0).
The change in entropy between two states (1 & 2) can therefore be obtained by integrating δQ/T along a reversible path that connects these states.

\mathrm {\triangle S_{sys}=S_2-S_1=\int_1^2{\left(\frac{\delta Q}T\right)}_{rev}}

Interpretation: The integration of δQ/T yields a property only for a reversible path, whereas for an irreversible path, it does not represent a property.

Hence, for any real process, the entropy change between two states can always be determined by integrating δQ/T along an imaginary internally reversible path between the specified states.

Entropy Change in Reversible Processes

Entropy change for reversible processes in a closed system is determined using the entropy balance equation with zero entropy generation.

Entropy Change for Reversible Adiabatic Process:

A reversible adiabatic process is one in which there is no heat transfer between the system and its surroundings. Since the process is reversible, there is no entropy generation:

\mathrm {\delta Q=0;\;\;\;S_{gen}=0}

From the definition of entropy change for a reversible process,

$\mathrm {\triangle S_{sys}=\int_1^2{\left(\frac{\cancel{\delta Q}}T\right)}_{rev}\Rightarrow\triangle S_{sys}=0}$

Thus, for a reversible adiabatic process, the entropy of the system remains constant.
Such a process is known as an isentropic process (iso = same, entropy = S).
Any process that is both adiabatic and reversible is referred to as an isentropic process.
The adiabatic operation of pumps, turbines, nozzles, and similar devices can be approximated as isentropic when frictional and irreversible effects are negligible.

Entropy Change for Reversible Isothermal Process:

A reversible isothermal process occurs reversibly at a constant temperature (T₀). For a reversible process, the change in entropy is defined as:

\mathrm {\triangle S_{sys}=\int_1^2{\left(\frac{\delta Q}T\right)}_{rev}}

Since the temperature remains constant (T = T₀) throughout the process, it can be taken outside the integral:

$\mathrm {\triangle S_{sys}=\frac1{T_0}\int_1^2{\left(\delta Q\right)}_{rev}=\frac{Q_{rev}}{T_0}}$

So, for a reversible isothermal process, the entropy change of the system depends on the direction of heat transfer. If the system receives heat, the entropy increases; if the system rejects heat, the entropy decreases.

Entropy Change in Irreversible Processes

For a closed system undergoing an irreversible process, the entropy change is determined using the entropy balance equation with a positive entropy generation term.

Entropy Change for an Irreversible Adiabatic Process:

An irreversible adiabatic process is one in which there is no heat transfer between the system and its surroundings:

\mathrm {\delta Q=0;\;\;\;S_{gen}>0}

The entropy balance for a closed system undergoing a process is given by:

$\mathrm {\triangle S_{sys}=\int_1^2\frac{\cancel{\delta Q_{actual}}}{T_{b,sys}}+S_{gen,sys}=S_{gen,sys}>0}$

Therefore, in an irreversible adiabatic process, entropy increases due to internal irreversibilities such as friction, viscosity, mixing, or turbulence.

Entropy Change of Irreversible Isothermal Process:

An irreversible isothermal process occurs irreversibly at a constant temperature (T₀). For an irreversible process, the entropy balance for a closed system is given by

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+S_{gen,sys}}

Since the temperature remains constant (T_b,sys = T₀) throughout the process, it can be taken outside the integral:

$\mathrm {\triangle S_{sys}=\frac1{T_0}\int_1^2\delta Q_{actual}+S_{gen,sys}=\frac{\delta Q_{actual}}{T_0}+S_{gen,sys}}$

So, for an irreversible isothermal process, the entropy change of the system depends on the direction of heat transfer and entropy generation. If the system receives heat, the entropy increases; if the system rejects heat, the entropy may decrease or increase depending on the entropy generation term.

Entropy Change of the Surroundings (Thermal Reservoirs)

The entropy change of the surroundings is evaluated to account for heat interactions with thermal reservoirs during thermodynamic processes.

Entropy Change of a Thermal Reservoir:

A thermal reservoir is an idealized system that can absorb or release a finite amount of heat without any change in its temperature (T_res).

Because the reservoir undergoes no internal gradients or finite driving forces, it is treated as an internally reversible system, and therefore,

\mathrm {S_{gen,res}=0}

The entropy balance for the thermal reservoir during a heat transfer process is

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{res}}{T_{b,res}}+\cancel{S_{gen,res}}}

Since the temperature of a thermal reservoir remains constant (T_b,res = T_res) throughout the process, it can be taken outside the integral:

$\mathrm {\triangle S_{sys}=\frac1{T_{res}}\int_1^2\delta Q_{res}=\frac{Q_{res}}{T_{res}}}$

The entropy change of a thermal reservoir depends only on the net heat it receives or releases, and not on the path of the heat transfer, due to its constant temperature.

Entropy Change Between Two Thermal Reservoirs:

Consider two thermal reservoirs: a hot reservoir at temperature T_H and a cold reservoir at temperature T_L, where (T_H > T_L). A finite quantity of heat Q is transferred from the hot reservoir to the cold reservoir.

The entropy change may be evaluated by defining either reservoir as the system and the other as the surroundings, or by treating both reservoirs together as a single isolated system.

For a thermal reservoir, the entropy change is given by

\mathrm {\triangle S_{sys}=\frac{Q_{res}}{T_{res}}}

Thus, the entropy changes of the hot and cold reservoirs are:

\mathrm {\triangle S_H=\frac{-Q}{T_H};\;\;\;\triangle S_L=\frac Q{T_L}}

The total entropy change of the combined (isolated) system is therefore

$\mathrm {\triangle S_{total}=\triangle S_H+\triangle S_L=\frac{-Q}{T_H}+\frac Q{T_L}=\left(\frac{T_H-T_L}{T_HT_L}\right)Q }$

Since T_H > T_L, it follows that

\mathrm {\triangle S_{total}>0}

Thus, the total entropy of the combined system always increases when heat is transferred across a finite temperature difference. This increase represents entropy generation due to external irreversibility (heat transfer irreversibility), even though each thermal reservoir is internally reversible.
Limiting Case: Reversible Heat Transfer

\mathrm {\triangle S_{total}=0\;\;\;only\;when\;T_H=T_L}

This corresponds to the idealized case of reversible heat transfer, in which heat is transferred across an infinitesimal temperature difference. In this limit, entropy generation vanishes, and the process becomes both internally and externally reversible.

Total Entropy Change (System + Surroundings):

The total entropy change of a thermodynamic process is obtained by summing the entropy changes of the system and its surroundings. The total entropy change of the universe is used to assess the reversibility or irreversibility of a process:

\mathrm {\triangle S_{total}=\triangle S_{sys}+\triangle S_{surr}}

Internally Reversible Process (Externally Irreversible):

In an internally reversible process, no irreversibilities occur within the system; therefore, the entropy generation within the system is zero $\mathrm {\left(S_{gen,sys}=0\right)}$

Entropy Change of the System: The entropy balance for the system is:

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+\cancel{S_{gen,sys}}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}}

The entropy change of the system may also be evaluated using a reversible heat transfer path between the same initial and final states:

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}}

Thus,

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}}

Entropy Change of the Surrounding: If the surrounding is modeled as a thermal reservoir at a constant boundary temperature (T_b,surr), the entropy change of the surroundings is

\mathrm {\triangle S_{surr}=\frac{Q_{surr}}{T_{b,surr}}}

Since heat lost by the system is gained by the surroundings (Q_surr = ‒Q_actual). Therefore,

\mathrm {\triangle S_{surr}=-\frac{Q_{actual}}{T_{b,surr}}}

Total Entropy Change: The total entropy change of the universe is

$\mathrm {\triangle S_{total}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}-\frac{Q_{actual}}{T_{b,surr}}}$

For an internally reversible but externally irreversible process, the overall entropy change of the universe is positive.

\mathrm {\triangle S_{total}>0}

Reversible Process (Internally + Externally):

In a fully reversible process, there are no irreversibilities within the system or in the surroundings. Heat transfer occurs across an infinitesimal temperature difference, so the system and surroundings boundary temperatures are effectively equal at all times: $\mathrm {\left(S_{gen}=0;\;\;T_{b,sys}=T_{b,surr}\right)}$

Entropy Change of the System: The entropy balance for the system is:

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+\cancel{S_{gen,sys}}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}}

The entropy change of the system may also be evaluated using a reversible heat transfer path between the same initial and final states:

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}}

Thus,

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}}

Entropy Change of the Surrounding: If the surrounding is modeled as a thermal reservoir at a constant boundary temperature (T_b,surr), the entropy change of the surroundings is

\mathrm {\triangle S_{surr}=\frac{Q_{surr}}{T_{b,surr}}}

Since heat lost by the system is gained by the surroundings (Q_surr = ‒Q_actual) and T_b,surr = T_b,sys. Therefore,

\mathrm {\triangle S_{surr}=-\frac{Q_{actual}}{T_{b,sys}}}

Total Entropy Change: The total entropy change of the universe is

$\mathrm {\triangle S_{total}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}-\frac{Q_{actual}}{T_{b,sys}}}$

For a reversible process, the overall entropy change of the universe is zero.

\mathrm {\triangle S_{total}=0}

Irreversible Process (Internally + Externally):

For an irreversible process, there are irreversibilities both within the system and at the system–surroundings interface: (S_gen,total > 0)

Entropy Change of the System: The entropy change of the system is given by the general entropy balance:

\triangle S_{sys}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+S_{gen,sys}

The entropy change of the system may also be evaluated using a reversible heat transfer path between the same initial and final states:

\mathrm {\triangle S_{sys}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}}

Thus,

\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+S_{gen,sys}=\int_1^2\frac{\delta Q_{rev}}{T_{b,sys}}

Entropy Change of the Surrounding: If the surrounding is modeled as a thermal reservoir at a constant boundary temperature (T_b,surr), the entropy change of the surroundings is

\mathrm {\triangle S_{surr}=\frac{Q_{surr}}{T_{b,surr}}}

Since heat lost by the system is gained by the surroundings (Q_surr = ‒Q_actual). Therefore,

\mathrm {\triangle S_{surr}=-\frac{Q_{actual}}{T_{b,surr}}}

Total Entropy Change: The total entropy change of the universe is

$\mathrm {\triangle S_{total}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+S_{gen,sys}-\frac{Q_{actual}}{T_{b,surr}}}$

For an irreversible process, the overall entropy change of the universe is positive.

\mathrm {\triangle S_{total}>0}

Entropy Change Due to Specific Processes

This section discusses the evaluation of entropy change for selected physical and chemical processes commonly encountered in closed systems, including phase change, mixing, real-substance behavior, and chemical reactions.

Reversible Phase Change:

Entropy changes occur during phase transitions such as fusion (melting), vaporization, and sublimation. Consider a pure substance undergoing a reversible phase change at constant temperature (T_tr) and pressure.

From the definition of entropy for a reversible process:

\triangle S_{phase}=\int_1^2\frac{\delta Q_{rev}}{T_b}

For a reversible process, the system boundary temperature is equal to the transition temperature (T_b = T_tr).
Since the temperature remains constant during the phase transition, it may be taken out of integration:

\triangle S_{phase}=\frac1{T_{tr}}\int_1^2\delta Q_{rev}=\frac{\delta Q_{rev}}{T_{tr}}

At constant pressure, the reversible heat absorbed or released during the phase transition is equal to the enthalpy change associated with the transition: (Q_rev = ∆H_phase)
Thus, the entropy change accompanying the phase transition is given by:

$\triangle S_{phase}=\frac{\triangle H_{phase}}{T_{tr}}$

Where ∆H_phase is the enthalpy change (latent heat) associated with the transition, and T_tr is the absolute equilibrium temperature at which the phase transition occurs.

Irreversible Phase Change:

For a pure substance undergoing an irreversible phase change at constant temperature and pressure, the entropy change of the system can still be evaluated by considering a hypothetical reversible path between the same initial and final equilibrium states:

\triangle S_{phase}=\frac{\triangle H_{phase}}{T_{tr}}

However, due to irreversibilities such as finite temperature difference (between the system boundary and the bulk phase-change region), non-equilibrium effects at the phase interface, or internal friction, entropy is generated within the system.
The entropy balance for the process is given by:

\triangle S_{phase}=\int_1^2\frac{\delta Q_{actual}}{T_{b,sys}}+S_{gen,sys}

Where T_b,sys is the temperature at the system boundary across which heat transfer occurs. For an irreversible process, boundary temperature and transition temperature may differ.

In general,

Q_{actual}\neq\triangle H_{phase};\;\;\;\;T_{b,sys}\neq T_{tr}

Q_{actual}\neq\triangle H_{phase};\;\;\;\;T_{b,sys}\neq T_{tr}