Second Law of Thermodynamics and Entropy Summary

Statistical Definition of Entropy and Microstates

• Macroscopic State Characterization: The state of a thermodynamic system is defined by its macroscopic state variables, such as:

  • $N$ (Number of molecules)

  • $V$ (Volume)

  • $U$ (Internal energy)

• Microstates ($\Omega$): There is a vast number of ways in which the values of macroscopic variables like $V$ and $U$ can be distributed among all the $N$ molecules. The specific number of these different ways is defined as the number of microstates (or microrealisations), denoted as $\Omega$, that correspond to the same single macroscopic state.

• Boltzmann’s Entropy Formula: For an insulated system at thermodynamic equilibrium, the entropy $S$ is statistically related to the number of microrealisations of the macroscopic state by the equation:

  • $S = k_B \tan(\Omega)$

  • Where $k_B$ is the Boltzmann constant.

Entropy as a State Function and the Thermodynamic Identity

• State Function Characteristics: Entropy is a state function. This implies that:

  • Its elementary variation $dS$ is an exact total differential.

  • The total biological variation between two states $A$ and $B$ is path-independent and calculated by: $\Delta S_{AB} = \int_A^B dS$.

• Entropy in Terms of Energy and Volume: Entropy is a measure of the ways internal energy ($U$) and volume ($V$) are distributed. It is thus a function of both: $S(U, V)$.

  • The differential for entropy is expressed as: $dS = \frac{\partial S}{\partial U} dU + \frac{\partial S}{\partial V} dV$.

• Intensive Variable Definitions: The intensive parameters Temperature ($T$) and Pressure ($P$) are defined through the partial derivatives of entropy:

  • Definition of Temperature: $\frac{1}{T} = \frac{\partial S}{\partial U}$

  • Definition of Pressure-to-Temperature ratio: $\frac{P}{T} = \frac{\partial S}{\partial V}$

• Thermodynamic Identity: Combining these definitions results in the fundamental relationship known as the thermodynamic identity:

  • $dS = \frac{1}{T} dU + \frac{P}{T} dV$

Physical Temperature and the Kelvin Scale

• Sign of Temperature: An increase or decrease in internal energy ($U$) at constant volume leads to a corresponding increase or decrease in the number of microstates $\Omega$, which subsequently implies an increase or decrease in entropy $S$.

• Constraint on $T$: Because of this direct relationship, the partial derivative $\frac{\partial S}{\partial U}$ (and thus the physical temperature) is necessarily positive, with a lower limit of absolute zero:

  • $\frac{1}{T} = \frac{\partial S}{\partial U} \geq 0$

• Kelvin Scale: This logic defines the Kelvin scale, where the zero point is established in a physically correct manner based on the minimum possible value of internal energy and microstate count.

Entropy Change during Reversible Processes

• General Reversible Variation: By inserting the First Law of Thermodynamics ($dU = \delta W + \delta Q$) into the thermodynamic identity, one obtains: $dS = \frac{\delta Q + \delta W}{T} + \frac{P dV}{T}$.

  • For any reversible process, the work is $\delta W = -P dV$, which simplifies the expression for the elementary change in entropy of a system to: $dS = \frac{\delta Q}{T}$.

  • The total change between state $A$ and $B$ is: $\Delta S_{AB} = \int_A^B \frac{\delta Q}{T}$.

• Reversible Adiabatic Process: In an adiabatic process, $\delta Q = 0$. For a reversible adiabatic process, this means $dS = 0$ and $\Delta S_{AB} = 0$. Such a process is described as being an isentropic process.

• Reversible Isochoric Process (Constant Volume):

  • The elementary heat received is $\delta Q = C_V dT$.

  • The elementary entropy change is $dS = \frac{C_V dT}{T}$.

  • The total variation is $\Delta S_{AB} = \int_{T_A}^{T_B} \frac{C_V dT}{T}$.

  • For an ideal gas: $\Delta S_{AB} = C_V \ln\left(\frac{T_B}{T_A}\right)$.

• Reversible Isobaric Process (Constant Pressure):

  • The elementary heat received is $\delta Q = C_P dT$.

  • The elementary entropy change is $dS = \frac{C_P dT}{T}$.

  • The total variation is $\Delta S_{AB} = \int_{T_A}^{T_B} \frac{C_P dT}{T}$.

  • For an ideal gas: $\Delta S_{AB} = C_P \ln\left(\frac{T_B}{T_A}\right)$.

• Reversible Isothermal Process (Constant Temperature):

  • Since $T$ is constant: $\Delta S_{AB} = \frac{1}{T} \int_A^B \delta Q = \frac{Q_{AB}}{T}$.

  • For an ideal gas, this is calculated as: $\Delta S_{AB} = nR \ln\left(\frac{V_B}{V_A}\right)$.

  • Useful algebraic relationships for rewriting this include: $\ln\left(\frac{V_B}{V_A}\right) = \ln\left(\frac{P_A}{P_B}\right)$.

  • Relationships for $nR$: $nR = C_P - C_V = C_V(\gamma - 1) = C_P\left(1 - \frac{1}{\gamma}\right)$.

• Sign Parity: Since $T \geq 0$, the change in entropy $\Delta S_{AB}$ during a reversible process always shares the same sign as the heat received by the system.

Entropy Balance: Exchanged and Created Entropy

• General Relationship: During any thermodynamic process (reversible or irreversible), the relationship between heat and entropy is generalized by the balance equation: $dS = \delta S_e + \delta S_c$.

  • $\delta S_e$ is the entropy exchanged with the external environment.

  • $\delta S_c$ is the entropy created within the system.

• Exchanged Entropy (\delta S_e): This is expressed as the heat received divided by the temperature of the external environment ($T_{ext}$): $\delta S_e = \frac{\delta Q}{T_{ext}}$.

  1. For a transition from $A$ to $B$ at a constant external temperature ($T_{ext} = \text{const}$):

     • $S_{e, AB} = \frac{1}{T_{ext}} Q_{AB} = \frac{\Delta U_{AB} - W_{AB}}{T_{ext}}$.

• Entropy Created (\delta S_c):

  1. Reversible Process: The system is in thermal equilibrium with the environment at every instant ($T_{ext} = T$). Thus, $\delta S_e = \frac{\delta Q}{T} = dS$. This implies no entropy is created: $\delta S_c = 0$ and $S_{c, AB} = 0$.

  2. Irreversible Process: Since entropy is a state function, the change $\Delta S_{AB}$ is the same as it would be for a reversible path between those states ($\Delta S_{AB, rev}$). The entropy created during an actual irreversible process is calculated as: $S_{c, AB} = \Delta S_{AB, rev} - S_{e, AB}$.

The Second Law of Thermodynamics and Engine Efficiency

• The Second Law Statement: During any physical process occurring in a thermodynamic system between initial state $A$ and final state $B$, the total entropy created is always non-negative:

  • $S_{c, AB} = \Delta S_{AB} - S_{e, AB} \geq 0$

  • $S_{c, AB} = 0$ only if the process is perfectly reversible.

• Consequences for Single Heat Source Systems: A system interacting with only a single heat source cannot convert heat completely into work over a cycle. During a cyclic process with one source, the system can only:

  1. Receive work (W > 0).

  2. Release heat (Q < 0).

• Requirements for Heat Engines: A heat engine is a device intended to release work (W < 0). Based on the Second Law, it must necessarily:

  • Be in contact with at least two parts of the external environment at different temperatures.

  • As a result of this necessity and the unavoidable creation of entropy (or simple energy transfer limits), a heat engine cannot have a thermal efficiency of 100%.