A perfect gas obeys the equation of state pV = nRT. The internal energy of a perfect gas remains unaffected by volume alterations. Therefore, the internal energy of a perfect gas is solely dependent on temperature.
Consider an ideal gas enclosed in a cylinder situated within a substantial constant-temperature bath. In an isothermal process, where the temperature remains constant, the change in internal energy equates to zero. Thus, according to the first law of thermodynamics, heat absorbed (q) equates to the negative work done (-w).
On increasing the pressure, the gas compresses gradually. The work exerted on it imparts energy to the gas, causing an infinitesimal increase in temperature. This infinitesimal temperature rise induces heat to flow out of the gas into the surrounding bath, effectively maintaining a constant gas temperature. Conversely, a decrease in pressure causes the gas to expand slowly and perform work on its surroundings. The resulting infinitesimal drop in gas temperature triggers heat to flow into the gas from the bath, maintaining a stable gas temperature.
Given the equation of state pV = nRT, at each stage, p = nRT/V, with V representing the volume at that specific stage of expansion or compression. Temperature remains constant in an isothermal expansion, allowing it (alongside n and R) to be taken outside the integral. Integrating the pressure-volume work using the equation of state provides the work of the reversible isothermal process of a perfect gas from the initial volume to the final volume at a given temperature.
In an expansion process where the final volume is greater than the initial volume, the work done on the gas is negative, and the heat added to the gas is positive. All the added heat translates into work performed by the gas, maintaining the internal energy constant for the perfect gas.
In an adiabatic process, where there's no heat transfer, the change in internal energy equals the work done. In adiabatic compression, the volume decreases, resulting in a positive change in internal energy and a corresponding rise in gas temperature. On the other hand, in an adiabatic expansion, the internal energy decreases, resulting in a corresponding drop in temperature.
An ideal gas obeys the equation pV = nRT, with its internal energy solely dependent on temperature.
To connect this molecular picture to experimental setups, consider the same gas contained in a cylinder with a movable piston placed within a constant-temperature bath.
Reducing the external pressure slowly expands the gas, slightly lowering the temperature. As a result, heat is transferred from the bath into the gas, maintaining a constant temperature.
In this reversible isothermal expansion, the internal energy change is zero, meaning heat equals work done.
Substituting the equation of state, which is the general relationship among pressure, volume, and temperature, into the pressure–volume work expression and integrating it relates the work done to the volume changes during an isothermal reversible expansion.
For adiabatic expansion, where no heat is transferred, the work done equals the internal energy change.
If expansion is reversible, equating work and internal energy and substituting the ideal gas relation for pressure, links the temperature and volume changes.
Integrating this equation and rearranging it relates the initial and final temperature and volumes of the gas.
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