Notes on Work Done by Gas and the First Law

Work Done by Gas on a Piston

  • When there is a moving part (e.g., a piston), there is an associated amount of work due to that motion.
  • This is the work done by the gas on its surroundings (the piston), not work done on the gas.
  • If the piston moves upward, the gas expands, and the gas does positive work on the piston.
  • The system in this context is the gas; the surroundings include the piston and external environment.

Key Concepts

  • Work by the gas is energy transfer due to a change in volume at some pressure.
  • Heat transfer q is energy added to (or removed from) the system via thermal interaction with the surroundings.
  • Experimental determination: both q and the work done by the gas (often denoted W or w) can be measured.
  • After measuring q and W, they are substituted into the governing energy balance to determine the internal energy change of the system.

Core Equations for Work and Heat

  • Work done by the system (gas) during a quasi-static process:
    W = \int P_{\text{ext}}\,dV
    If the process is quasi-static and the internal pressure equals the external pressure, this can be written as
    W = \int P\,dV.

  • For a constant external pressure (isobaric process):
    W = P{\text{ext}}\,\Delta V = P{\text{ext}}\,(Vf - Vi).

  • Sign convention: W > 0 when the system does work on the surroundings (e.g., piston expands).

  • Heat transferred to the system: q\,. (Positive q means heat flows into the gas.)

  • First Law of Thermodynamics (energy balance):

    • Using the convention where W is the work done by the system:
      \Delta U = q - W.
    • If a different convention is used where w is the work done on the system (opposite sign to W):
      \Delta U = q + w.
    • These two forms are equivalent provided the sign conventions for W and w are used consistently.

Experimental Considerations

  • Measuring q: calorimetry (temperature change, heat capacity, calorimeter constants).
  • Measuring W: can be inferred from pressure-volume data or external pressure and volume change; for isobaric or quasi-static processes, this simplifies to W = P_ext ΔV.
  • Practical note: real processes may involve non-quasi-static flow, friction, and heat losses, which affect the measured q and W.

Substitution into the Energy Balance

  • After determining q and W (or w), substitute into the energy balance to obtain the internal energy change:
    \Delta U = q - W
    (or \Delta U = q + w\$ depending on the chosen sign convention).
  • This yields the net change in the gas's internal energy during the process.

Conceptual Understanding

  • Upward piston movement corresponds to expansion work by the gas; the system loses/internal energy can be converted into work on the surroundings.
  • The energy conservation perspective connects microscopic molecular interactions to macroscopic observables (q, W, and (\Delta U)).

Connections to Foundational Principles

  • First Law of Thermodynamics: energy conservation across heat, work, and internal energy.
  • Relationship between pressure, volume, and energy in standing gas systems undergoing piston movements.

Example (Illustrative)

  • Suppose a gas expands at constant external pressure (P{\text{ext}}) from (Vi) to (V_f) with measured q = 300 J and W = 120 J (work done by the gas).
    • Then:
      \Delta U = q - W = 300\ \text{J} - 120\ \text{J} = 180\ \text{J}.
  • If a different convention is used (where w is work done on the system, so (W = -w)):
    • Then (\Delta U = q + w = q - W) as well, yielding the same (\Delta U) when consistently applying signs.

Real-World Relevance and Practical Implications

  • In engines, refrigerators, and many industrial processes, determining q and W is essential to evaluate efficiency and performance.
  • Isothermal, isobaric, and adiabatic processes illustrate different balances of heat and work and their impact on (\Delta U).
  • Experimental challenges include accurately measuring heat transfer, accounting for non-ideal gas behavior, and minimizing heat losses to surroundings.

Summary

  • The work done by the gas on a piston during volume change is given by:
    W = \int P{\text{ext}}\,dV, with special case for constant pressure: W = P{\text{ext}}\Delta V.
  • Heat added to the gas is q, and the first law relates these quantities to the change in internal energy:
    \Delta U = q - W (or, with the alternate convention, \Delta U = q + w\$).
  • Practically, you measure q and W experimentally and substitute into the energy balance to obtain (\Delta U) for the process.