Introduction to Thermodynamics: Gas Work

Relevance of the Topic

Thermodynamics, more specifically gas work, constitutes one of the fundamental pillars of Physics. This theory allows the understanding of the macroscopic aspects of the behavior of matter, which is of great utility not only in Physics but also in various areas of science and engineering. The ability to quantify the work done by a gas is critical to understand the principles of operation of engines, heat pumps, and other devices essential to modern technology. Furthermore, the concept of work in thermodynamics helps us understand how energy can be transferred and transformed between different forms, deepening our understanding of the universe around us.

Contextualization

Within the broad topic of Thermodynamics, the discussion on the work done by a gas occupies a central place. This concept is a natural continuation of discussions on the pressure and volume of a gas, forming an inseparable trinomial within Thermodynamics. Through the study of this theme, we are able to connect previous concepts - such as Boyle-Mariotte's Law and Charles's Law - with a more concrete and tangible result: the performance of work by the gas. This creates a complete framework of gas dynamics, providing a deeper understanding of the physical world and its technological applications. Gas work is therefore an essential component of the Physics curriculum in High School, serving as a conceptual basis for understanding more complex phenomena that will be explored later, such as energy conservation and the study of thermodynamic processes in general.

Lecture Note - Thermodynamics: Gas Work

Theoretical Development

• Concept of Thermodynamic Work:

  • The work done by a gas is a form of energy transfer, in which the force applied to the gas results in a displacement of the gas.

  • In the thermodynamic context, work is defined as the product of the pressure (P) applied to the gas and the displacement (ΔV) that the gas undergoes, in the direction of the applied force. That is, W = P * ΔV.

  • The value of thermodynamic work is directly dependent on the change in volume of the gas and the pressure to which the gas is subjected.

• Calculation of Gas Work:

  • For ideal gases, the calculation of work is relatively simple. If the gas expands (or is compressed) from an initial volume Vi to a final volume Vf, at a constant pressure P, the work can be calculated as W = P * (Vf - Vi).

  • However, when the gas pressure varies during expansion (or compression), it is necessary to integrate the work expression over the volume variation to find the correct value. The work in an adiabatic process, for example, is given by W = (P2 * V2 - P1 * V1) / (1 - γ), where γ is the adiabatic index, a gas property.

• Reversible and Irreversible Processes:

  • An important distinction to make is between reversible and irreversible processes. In a reversible process, the system and its surroundings can be brought back to their initial state with the transfer of an infinitesimal amount of work. In an irreversible process, this is not possible.

  • In a reversible process, the work is maximum and is given by the force moment by the distance, in cases of isothermal expansion or compression, this results in W = (1.38 * 10^-23) * T * ln(Vf/Vi), where T is the gas temperature.

  • In an irreversible process, the work is less than in the corresponding reversible process. This occurs because part of the system's energy is dissipated as thermal energy.

Key Terms

• Thermodynamic Work: The energy transferred to or from a thermodynamic system as a result of a force being applied to it during a straight-line displacement in the direction of the force.

• Ideal Gas: A theoretical model that describes a gas where particles do not interact with each other, except during perfectly elastic collisions.

• Reversible Process: A thermodynamic process in which the system and its surroundings can be brought back to their initial states through an infinitesimally similar path.

• Irreversible Process: A thermodynamic process that cannot be completely reversed to its initial state by an infinitesimally similar path.

Examples and Cases

• Example 1: Isobaric Compression:

  • Imagine a gas contained in a cylinder, under a constant pressure of 2 atmospheres. The cylinder is then compressed from an initial volume of 5 liters to a final volume of 3 liters. The work done on the gas can be calculated as W = P * (Vf - Vi) = 2 atm * (3 L - 5 L) = -4 atm L. The negative sign indicates that work is done on the gas, as opposed to work being done by it (expansion).

• Example 2: Reversible Isothermal Expansion of an Ideal Gas:

  • An ideal gas at 300 K is contained in a volume of 4 liters. It is then reversibly expanded to a final volume of 8 liters. The work done by the gas can be calculated using W = (1.38 * 10^-23) * T * ln(Vf/Vi) = (1.38 * 10^-23) * 300 K * ln(8/4) = 8.617 * 10^-21 J.

• Example 3: Irreversible Adiabatic Expansion of an Ideal Gas:

  • In an irreversible adiabatic process, the work of an ideal gas is less than in the corresponding reversible process. For example, if the gas from the previous example were to expand irreversibly to the same final volume of 8 liters, but without exchanging heat with the surroundings, the work done would be less.

Detailed Summary

• Key Points:

  • The concept of thermodynamic work is crucial to understanding gas thermodynamics. Work is the result of the force applied to the gas causing a displacement of the gas.

  • The general formula to calculate thermodynamic work is W = P * ΔV, where P is the pressure and ΔV is the change in gas volume. However, in situations where pressure is not constant, more complex expressions are needed to calculate work.

  • The distinction between reversible and irreversible processes is of vital importance. The work in an irreversible process is always less than in the corresponding reversible process, due to the dissipation of energy as thermal energy.

• Conclusions:

  • The study of gas work has allowed us to connect the pressure and volume of a gas with energy transfer, consolidating our understanding of the fundamentals of thermodynamics.

  • Reversible thermodynamic processes are theoretically ideal, while irreversible processes are more common in practice and result in less work due to the dissipation of energy.

  • Gas work is not just a theoretical concept; it has practical applications in a variety of fields, especially in engineering and industry.

• Exercises:

  1. Calculate the work done by a gas that expands from 5 liters to 15 liters, under a constant pressure of 2 atmospheres.

  2. In an experiment, a gas contained in an 8-liter cylinder is compressed to 4 liters. However, during the compression process, the gas pressure increases linearly from 2 atmospheres to 8 atmospheres. Determine the work done on the gas.

  3. In an adiabatic process, an ideal gas contained in a cylinder is compressed from an initial volume of 6 liters to a final volume of 2 liters. The initial gas pressure was 5 atmospheres and the adiabatic index γ for the gas is 1.4. Calculate the work done by the gas.