6.1 Entropy changes in various processes

Entropy changes in thermodynamic processes are crucial for understanding how systems behave. These changes depend on factors like temperature, pressure, and volume, and can be calculated using specific formulas for different types of processes.

Entropy is a state function, meaning its change only depends on the initial and final states, not the path taken. This concept is fundamental to the second law of thermodynamics, which states that the entropy of an isolated system always increases or remains constant.

Entropy Changes in Thermodynamic Processes

Entropy changes in thermodynamic processes

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• Isothermal process maintains constant temperature while entropy change is calculated using $\Delta S = nR \ln \frac{V_2}{V_1} = nR \ln \frac{P_1}{P_2}$
  • Entropy increases when volume increases or pressure decreases (gas expansion)
  • Entropy decreases when volume decreases or pressure increases (gas compression)
• Isobaric process maintains constant pressure while entropy change is calculated using $\Delta S = nC_p \ln \frac{T_2}{T_1}$
  • Entropy increases when temperature increases (heating)
  • Entropy decreases when temperature decreases (cooling)
• Isochoric process maintains constant volume while entropy change is calculated using $\Delta S = nC_v \ln \frac{T_2}{T_1}$
  • Entropy increases when temperature increases (heating at constant volume)
  • Entropy decreases when temperature decreases (cooling at constant volume)
• Adiabatic process involves no heat transfer and entropy change is $\Delta S = 0$ for a reversible adiabatic process
  • Reversible adiabatic processes (isentropic) maintain constant entropy
  • Irreversible adiabatic processes (non-isentropic) result in entropy increase

Entropy change of ideal gases

• For a reversible process, the entropy change of an ideal gas is calculated using $\Delta S = nC_v \ln \frac{T_2}{T_1} + nR \ln \frac{V_2}{V_1}$
  • Accounts for entropy changes due to both temperature and volume changes
  • Specific heat at constant volume ($C_v$) used for temperature-related entropy change
  • Gas constant ($R$) used for volume-related entropy change
• Entropy change depends on initial and final states, not the path taken between them (state function)
  • Different reversible processes between the same initial and final states yield the same entropy change

System and surroundings entropy analysis

• Total entropy change of the universe ($\Delta S_{universe}$) is the sum of entropy changes in the system ($\Delta S_{system}$) and surroundings ($\Delta S_{surroundings}$)
  • $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$
• For reversible processes:
  1. $\Delta S_{universe} = 0$
  2. $\Delta S_{system} = -\Delta S_{surroundings}$
• For irreversible processes:
  1. $\Delta S_{universe} > 0$
  2. Entropy of the universe always increases
• Analyzing entropy changes in both system and surroundings provides a comprehensive understanding of the process

Entropy as a state function

• Entropy is a state function, meaning its value depends only on the current state, not the path taken to reach that state
  • Change in entropy ($\Delta S$) between two states is independent of the process path
• Implications of entropy as a state function:
  1. Entropy change for a cyclic process is always zero ($\oint dS = 0$)
     • System returns to its initial state after a complete cycle
  2. Entropy change for a reversible process between two states is the same as any other reversible process between the same states
     • Enables calculation of entropy changes using any convenient reversible path

Entropy and the Second Law of Thermodynamics

Relationship between entropy and the second law

• Second law of thermodynamics states that entropy of an isolated system always increases or remains constant over time
  • For spontaneous processes in isolated systems, $\Delta S_{universe} > 0$
  • For reversible processes in isolated systems, $\Delta S_{universe} = 0$
• Second law provides a direction for spontaneous processes and establishes irreversibility
  • Heat spontaneously flows from hot to cold objects, never the reverse without external work (thermal equilibrium)
  • Entropy of the universe increases in all spontaneous processes (arrow of time)