Glossary

13.3 First and second-law analysis of gas mixtures

6 min read•july 30, 2024

Gas mixtures are a crucial part of thermodynamics. The first law helps us understand energy changes in these mixtures, while the second law deals with and process reversibility. These concepts are key to analyzing real-world systems involving multiple gases.

Applying these laws to gas mixtures lets us calculate work, , and internal energy changes. We can also determine entropy changes and assess process reversibility. This knowledge is essential for designing efficient systems and understanding natural phenomena involving gas mixtures.

Thermodynamics of Ideal Gas Mixtures

First Law of Thermodynamics for Ideal Gas Mixtures

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  • The states that the change in internal energy of a system equals the heat added to the system minus the work done by the system
  • For an , the internal energy depends only on temperature and can be calculated using the specific heats of the individual components
  • The work done by an ideal gas mixture during a process can be determined using the ideal gas equation of state and the process path
  • The heat transfer during a process involving an ideal gas mixture can be calculated using the first law of thermodynamics, the change in internal energy, and the work done
  • The first law of thermodynamics applies to various processes involving ideal gas mixtures (isothermal, isobaric, isochoric, and adiabatic processes)

Analyzing Processes with Ideal Gas Mixtures

  • Apply the first law of thermodynamics to analyze processes involving ideal gas mixtures
    • Determine the change in internal energy using the specific heats and temperature change
    • Calculate the work done using the pressure-volume relationship and the process path
    • Evaluate the heat transfer using the first law, change in internal energy, and work done
  • Examples of processes involving ideal gas mixtures:
    • : constant temperature, heat transfer equals work done
    • Isobaric process: constant pressure, work done equals pΔVpΔV
    • Isochoric process: constant volume, no work done, heat transfer equals change in internal energy
    • : no heat exchange with surroundings, work done calculated using adiabatic process equation

Work, Heat, and Internal Energy Changes

Calculating Work Done by Ideal Gas Mixtures

  • The work done by an ideal gas mixture during a process can be calculated by integrating the pressure-volume relationship along the process path
    • For a reversible isothermal process, work done equals nRTln(V2/V1)nRTln(V_2/V_1)
    • For a reversible isobaric process, work done equals p(V2−V1)p(V_2-V_1)
    • For an isochoric process, work done is zero due to no change in volume
    • For a reversible adiabatic process, work done can be calculated using the adiabatic process equation and the specific heat ratio of the gas mixture
  • Examples of work calculations for ideal gas mixtures:
    • Isothermal expansion of a gas mixture from 1 L to 2 L at 300 K
    • Isobaric compression of a gas mixture from 2 L to 1 L at 1 atm
    • Isochoric heating of a gas mixture at constant volume
    • Adiabatic expansion of a gas mixture with a specific heat ratio of 1.4

Determining Internal Energy Changes and Heat Transfer

  • The change in internal energy of an ideal gas mixture can be determined using the specific heats of the individual components and the change in temperature
  • The heat transfer during a process involving an ideal gas mixture can be calculated using the first law of thermodynamics, the change in internal energy, and the work done
    • For an isothermal process, heat transfer equals work done because internal energy remains constant
    • For an isobaric process, heat transfer equals mcp(T2−T1)mc_p(T_2-T_1)
    • For an isochoric process, heat transfer equals change in internal energy because no work is done
    • For an adiabatic process, heat transfer is zero due to no heat exchange with the surroundings
  • Examples of internal energy and heat transfer calculations:
    • Isothermal compression of a gas mixture, determining heat transfer
    • Isobaric heating of a gas mixture, calculating change in internal energy and heat transfer
    • Isochoric cooling of a gas mixture, evaluating change in internal energy and heat transfer
    • Adiabatic compression of a gas mixture, analyzing change in internal energy and work done

Entropy Changes and Second Law Implications

Entropy Changes in Ideal Gas Mixtures

  • The introduces entropy, a measure of the disorder or randomness of a system
  • The change in entropy of an ideal gas mixture during a process can be calculated using the specific heats of the individual components and the changes in temperature and pressure
    • For a reversible isothermal process, entropy change equals heat transfer divided by constant temperature
    • For a reversible isobaric process, entropy change can be calculated using specific heat at constant pressure and ln(T2/T1)ln(T_2/T_1)
    • For a reversible isochoric process, entropy change can be calculated using specific heat at constant volume and ln(T2/T1)ln(T_2/T_1)
    • For a reversible adiabatic process, entropy change is zero due to no heat transfer
  • Examples of entropy change calculations for ideal gas mixtures:
    • Reversible isothermal expansion of a gas mixture, determining entropy change
    • Reversible isobaric cooling of a gas mixture, calculating entropy change
    • Reversible isochoric heating of a gas mixture, evaluating entropy change
    • Reversible adiabatic compression of a gas mixture, analyzing entropy change

Second Law Analysis of Ideal Gas Mixture Processes

  • The second law of thermodynamics states that the total entropy of an isolated system always increases for irreversible processes and remains constant for reversible processes
  • The entropy generation during a process involving an ideal gas mixture can be used to evaluate the irreversibility of the process
    • For a reversible process, entropy generation is zero, indicating no dissipative effects
    • For an irreversible process, entropy generation is positive, indicating the presence of dissipative effects
  • Examples of second law analysis for ideal gas mixture processes:
    • Analyzing the entropy generation during the mixing of two different ideal gases
    • Evaluating the irreversibility of a throttling process (unrestrained expansion) of a gas mixture
    • Comparing the entropy changes in reversible and irreversible adiabatic expansions of a gas mixture

Reversibility vs Irreversibility of Processes

Reversible Processes

  • A reversible process can be reversed without leaving any trace on the surroundings and occurs infinitely slowly through a series of equilibrium states
  • The reversibility of a process involving an ideal gas mixture can be determined by analyzing the entropy generation during the process
    • For a reversible process, entropy generation is zero, indicating no dissipative effects (friction, heat transfer through a finite temperature difference, or unrestrained expansion)
  • The reversibility of a process can also be evaluated by examining the process path and comparing it with the corresponding reversible process
  • Examples of reversible processes involving ideal gas mixtures:
    • Reversible isothermal compression or expansion
    • Reversible isobaric heating or cooling
    • Reversible isochoric heating or cooling
    • Reversible adiabatic compression or expansion

Irreversible Processes

  • An irreversible process cannot be reversed without leaving a trace on the surroundings and occurs at a finite rate
  • The irreversibility of a process involving an ideal gas mixture can be quantified using the concept of exergy destruction, which represents the lost potential to do useful work due to entropy generation
  • Examples of irreversible processes involving ideal gas mixtures:
    • Throttling process (unrestrained expansion) of a gas mixture
    • Mixing of two different ideal gases at different temperatures or pressures
    • Heat transfer through a finite temperature difference between a gas mixture and its surroundings
    • Friction during the flow of a gas mixture through pipes or ducts
  • The irreversibility of a process can be reduced by minimizing entropy generation through the use of efficient designs, heat exchangers, and insulation

Key Terms to Review (17)

Adiabatic process: An adiabatic process is a thermodynamic process in which no heat is transferred into or out of the system. During this type of process, any change in the internal energy of the system is solely due to work done on or by the system, making it essential in understanding how systems behave under different conditions.
Carnot Cycle: The Carnot cycle is an idealized thermodynamic cycle that represents the most efficient possible heat engine operating between two temperature reservoirs. It provides a standard for measuring the performance of real engines and illustrates the principles of energy transfer, work, and heat efficiency in thermodynamic processes.
Clausius Inequality: The Clausius Inequality is a fundamental principle in thermodynamics that states that for any real process, the change in entropy of a system is greater than or equal to the heat transferred into the system divided by the temperature at which the transfer occurs. This inequality helps establish the direction of thermodynamic processes and emphasizes that real processes are irreversible, highlighting the importance of entropy in understanding energy transformations.
Combustion analysis: Combustion analysis is a technique used to determine the elemental composition of a substance by measuring the amounts of products formed during its combustion. This method typically involves burning a sample in excess oxygen and analyzing the resulting gases, such as carbon dioxide and water vapor, to ascertain the amounts of carbon, hydrogen, and sometimes nitrogen present. This analysis plays a significant role in understanding the behavior of gas mixtures and their energy content.
Enthalpy: Enthalpy is a thermodynamic property defined as the sum of a system's internal energy and the product of its pressure and volume, represented by the equation $$H = U + PV$$. This concept is crucial for understanding energy transfer in processes involving heat and work, especially in closed systems, where enthalpy changes can indicate how much energy is absorbed or released during physical and chemical transformations.
Entropy: Entropy is a measure of the disorder or randomness in a system, reflecting the degree of energy dispersal at a specific temperature. It connects to fundamental concepts like the direction of processes, equilibrium states, and the efficiency of energy transformations in various thermodynamic cycles.
First Law of Thermodynamics: The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed from one form to another, which means the total energy of an isolated system remains constant. This principle underlies various processes, cycles, and energy interactions that involve heat, work, and mass transfer in different systems.
Heat Transfer: Heat transfer is the process of energy moving from a warmer object to a cooler one due to a temperature difference. This phenomenon plays a crucial role in various thermodynamic processes, affecting how systems interact with their surroundings and how energy is conserved or transformed within them.
Ideal gas mixture: An ideal gas mixture is a collection of multiple gases that behave independently of one another while occupying the same volume and maintaining uniform temperature and pressure conditions. Each component in the mixture follows the ideal gas law, and the overall behavior of the mixture can be analyzed using simple mathematical relationships, which make it easier to calculate properties like density, pressure, and temperature of the mixture.
Isothermal process: An isothermal process is a thermodynamic process in which the temperature of a system remains constant while the system undergoes a change in volume or pressure. This type of process is crucial for understanding how systems interact with their surroundings and how energy is exchanged in various thermodynamic cycles.
Molar fraction: Molar fraction is the ratio of the number of moles of a specific component in a mixture to the total number of moles of all components in that mixture. This concept is essential in understanding gas mixtures, as it helps describe the composition and behavior of gases under various conditions, particularly when applying first and second-law analyses to these mixtures.
Partial pressure: Partial pressure is the pressure exerted by a single component of a gas mixture when it occupies the same volume as the entire mixture. This concept is essential for understanding how different gases behave when mixed, as each gas contributes to the total pressure based on its own properties and concentration. The idea of partial pressure helps in analyzing gas mixtures, calculating properties of ideal gases, and applying the first and second laws of thermodynamics to systems containing multiple gases.
Rankine cycle: The Rankine cycle is a thermodynamic cycle that converts heat into work through a series of processes involving a working fluid, typically water or steam. It consists of four main processes: isentropic compression, isobaric heat addition, isentropic expansion, and isobaric heat rejection, making it a foundational concept in the study of heat engines and energy conversion systems.
Real gas mixture: A real gas mixture refers to a combination of different gases that do not behave ideally under all conditions, typically due to interactions between gas molecules and variations in pressure and temperature. Unlike ideal gas mixtures, real gas mixtures take into account factors such as non-ideal behavior and the physical properties of each component, which can significantly affect the overall thermodynamic characteristics of the mixture.
Refrigeration cycle: The refrigeration cycle is a thermodynamic process that removes heat from a designated area to lower its temperature, typically using a refrigerant. This cycle involves a series of phase changes and energy transfers that allow heat to be absorbed from the surroundings and expelled elsewhere, making it essential for various applications such as cooling systems and heat pumps.
Second Law of Thermodynamics: The Second Law of Thermodynamics states that the total entropy of an isolated system can never decrease over time, and it tends to increase, leading to the concept that energy transformations are not 100% efficient. This law establishes the directionality of processes, implying that certain processes are irreversible and energy has a quality that degrades over time, connecting tightly to concepts of heat transfer, work, and system analysis.
Thermal efficiency: Thermal efficiency is a measure of how well an energy conversion system, such as a heat engine, converts heat energy into useful work. It is defined as the ratio of the useful work output to the heat input, typically expressed as a percentage. This concept is crucial for evaluating and optimizing the performance of various thermodynamic cycles and systems.
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