Thermodynamic Processes Types
Isothermal Processes
An isothermal process occurs at constant temperature. For an ideal gas, this means PV = constant (Boyle law), so the PV diagram shows a hyperbola. The internal energy of an ideal gas depends only on temperature, so during an isothermal process, delta U = 0 and the first law gives Q = W: all heat absorbed is converted to work (or all work done on the gas is released as heat).
Isothermal processes require the system to be in thermal contact with a large heat reservoir that maintains the temperature. The process must also be slow enough for the gas to equilibrate with the reservoir at each step. In practice, processes approach isothermal behavior when they occur slowly in well-insulated equipment with large thermal mass, or when heat transfer is efficient.
The work done during an isothermal expansion of an ideal gas from V{sub}1{/sub} to V{sub}2{/sub} is W = nRT ln(V{sub}2{/sub}/V{sub}1{/sub}). This is the maximum work obtainable for an expansion at fixed temperature, and it is larger than the work for any adiabatic expansion between the same volumes because the heat input keeps the pressure higher throughout the expansion.
Adiabatic Processes
An adiabatic process involves no heat exchange with the surroundings (Q = 0). The first law reduces to dU = -W: any work done by the gas comes entirely from its internal energy, causing the temperature to drop. Conversely, compressing a gas adiabatically raises its temperature because the work input increases the internal energy.
For a reversible adiabatic process with an ideal gas, PVgamma = constant, where gamma = C{sub}p{/sub}/C{sub}v{/sub} is the heat capacity ratio (about 1.4 for diatomic gases like air). The adiabatic curve on a PV diagram is steeper than the isothermal curve, reflecting the fact that temperature changes during the process. The work is W = (P{sub}1{/sub}V{sub}1{/sub} - P{sub}2{/sub}V{sub}2{/sub})/(gamma - 1).
Adiabatic processes are good approximations for rapid processes where there is not enough time for significant heat transfer. The compression stroke in an internal combustion engine, the expansion of gas through a turbine nozzle, and the rise of air parcels in the atmosphere are all approximately adiabatic. The adiabatic lapse rate (the rate at which air temperature drops with altitude) is about 9.8 degrees Celsius per kilometer for dry air, a direct consequence of adiabatic expansion as air pressure decreases with height.
Isobaric and Isochoric Processes
An isobaric process occurs at constant pressure. On a PV diagram, it appears as a horizontal line. The work done is simply W = P delta V. The heat transferred equals the change in enthalpy: Q = delta H = nC{sub}p{/sub} delta T for an ideal gas. Heating water in an open pot at atmospheric pressure, or the intake and exhaust strokes of an engine at atmospheric pressure, approximate isobaric processes.
An isochoric (or isometric) process occurs at constant volume. No PV work is done because dV = 0. All heat added goes directly into changing the internal energy: Q = delta U = nC{sub}v{/sub} delta T for an ideal gas. Heating a sealed, rigid container is an isochoric process. The combustion of fuel in a rigid bomb calorimeter is approximately isochoric, which is why bomb calorimetry measures the internal energy of combustion rather than the enthalpy.
The relationship between C{sub}p{/sub} and C{sub}v{/sub} for an ideal gas is C{sub}p{/sub} - C{sub}v{/sub} = R. The heat capacity at constant pressure is always larger than at constant volume because part of the heat input at constant pressure goes into doing expansion work rather than raising the temperature. This difference is exactly R (the gas constant) per mole for ideal gases, and slightly larger for real gases.
Each process type constrains the system differently, determining the split between heat and work. Identifying the correct process type is the first step in any thermodynamic calculation.
Polytropic Processes and Real Systems
The four fundamental process types are special cases of the polytropic process PVn = constant, where n is the polytropic index. For n = 0, the process is isobaric. For n = 1, it is isothermal. For n = gamma, it is adiabatic. For n approaching infinity, it is isochoric. Real processes often fall between these limits and can be described by an intermediate value of n determined from experimental data.
In engineering practice, compressors and expanders often operate between isothermal and adiabatic limits. A slow, well-cooled compressor approaches isothermal compression (minimum work input), while a fast, insulated compressor approaches adiabatic compression. The actual compression follows a polytropic path with an index between 1 and gamma. Multistage compression with intercooling approaches the isothermal limit more closely by removing heat between stages.
Real processes are always irreversible to some degree, involving friction, turbulence, non-uniform temperatures, and finite-rate heat transfer. These irreversibilities generate entropy and reduce the useful work output (or increase the required work input) compared to the reversible idealization. The gap between real and ideal performance is quantified by isentropic efficiency for adiabatic devices and by second-law efficiency for general processes.