Adiabatic Process: Work Done & Relations
Exploring thermodynamic processes where no heat enters or leaves the system.
An Adiabatic Process is a thermodynamic process in which there is no exchange of heat between the system and the surroundings. This implies that the system is thermally insulated.
Mathematical Condition:
From the First Law of Thermodynamics ($\Delta U = q + w$):
Work is done at the expense of Internal Energy.
1. Thermodynamic Relations (Poisson's Equations)
For a Reversible Adiabatic Process involving an ideal gas, the state variables ($P, V, T$) are related by the following equations, involving the adiabatic index $\gamma$ (Gamma), where $\gamma = C_p / C_v$.
1. Relation between P and V
2. Relation between T and V
3. Relation between P and T
2. Slope of Adiabatic Curve
On a PV diagram, the slope of an adiabatic process is steeper than that of an isothermal process.
- Isothermal Slope: $\frac{dP}{dV} = -\frac{P}{V}$
- Adiabatic Slope: $\frac{dP}{dV} = -\gamma \frac{P}{V}$
Since $\gamma > 1$, the adiabatic curve is always steeper.
3. Work Done in Reversible Adiabatic Process
The work done ($w$) is given by $\int -P_{ext} dV$. Using the relation $PV^\gamma = K$:
Formula in terms of P and V:
(Using IUPAC sign convention where w is work done ON the system. If calculating work done BY gas, sign is reversed)
Substituting $PV = nRT$:
Formula in terms of Temperature:
Since $C_v = \frac{R}{\gamma - 1}$, this simplifies to:
Key Observations:
- Adiabatic Expansion: System does work ($w < 0$). Internal energy decreases ($\Delta U < 0$). Temperature falls (Cooling).
- Adiabatic Compression: Work is done on system ($w > 0$). Internal energy increases ($\Delta U > 0$). Temperature rises (Heating).
Knowledge Check
Test your understanding of Adiabatic Processes
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