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Part of THERM-01 — Thermodynamics & Kinetic Theory of Gases

Comparison Note: Four Thermodynamic Processes

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Complete Process Comparison Table

Property	Isothermal	Adiabatic	Isochoric	Isobaric
Constant quantity	Temperature (T)	No heat exchange (Q = 0)	Volume (V)	Pressure (P)
What is zero?	$\Delta U$ = 0	Q = 0	W = 0	Nothing is zero
Q formula	nRT ln( $V_{2}$ / $V_{1}$ )	0	nCᵥ $\Delta T$	nCₚ $\Delta T$
W formula	nRT ln( $V_{2}$ / $V_{1}$ )	nCᵥ( $T_{1}$ − $T_{2}$ )	0	P $\Delta V$ = nR $\Delta T$
$\Delta U$ formula	0	nCᵥ( $T_{2}$ − $T_{1}$ )	nCᵥ $\Delta T$	nCᵥ $\Delta T$
Governing equation	PV = const	PV^γ = const	P/T = const	V/T = const
PV curve shape	Rectangular hyperbola	Steeper hyperbola	Vertical line	Horizontal line
Temperature changes?	No	Yes	Yes	Yes
Applied condition	Slow process with thermostat	Sudden/fast process, insulated	Rigid container	Open cylinder (constant pressure)

Key Comparisons

Isothermal vs Adiabatic:

Both can be expansion processes; adiabatic is always steeper on PV diagram.
Isothermal: temperature stays constant because heat flows in/out.
Adiabatic: no heat flow, so the gas must cool on expansion to do work.

Isochoric vs Isobaric:

Isochoric: volume fixed, all energy becomes internal. No work possible.
Isobaric: pressure fixed, work is done. Extra heat is needed compared to isochoric for the same $\Delta T$ (this is why Cₚ > Cᵥ).

Why Cₚ > Cᵥ always: At constant pressure, gas must also do work of expansion (W = nR $\Delta T$ ). So Cₚ = Cᵥ + R (Mayer's relation).

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