Isothermal Process: Work Done & Relations
Understanding thermodynamic changes at constant temperature.
An Isothermal Process is a thermodynamic process in which the temperature of the system remains constant throughout ($dT = 0$). For this to happen, the system must exchange heat with the surroundings to compensate for work done.
1. Conditions and State Variables
Since temperature is constant ($T = \text{const}$), for an Ideal Gas, the internal energy depends only on temperature.
Key Properties (Ideal Gas):
Consequently, Enthalpy change $\Delta H = 0$ as well.
2. First Law of Thermodynamics
According to the First Law: $\Delta U = q + w$.
Substituting $\Delta U = 0$:
Heat absorbed by the system ($q > 0$) is entirely used to do work of expansion ($w < 0$).
3. PV Relation (Boyle's Law)
For an ideal gas at constant temperature, the ideal gas equation $PV = nRT$ becomes:
The PV graph is a rectangular hyperbola. Its slope is less steep than an adiabatic curve.
4. Work Done Calculations
A. Reversible Isothermal Expansion
When the process occurs infinitely slowly, keeping pressure in equilibrium ($P_{ext} \approx P_{gas}$). $$ w = -\int_{V_1}^{V_2} P \, dV = -\int_{V_1}^{V_2} \frac{nRT}{V} \, dV $$
Work Done (Reversible)
B. Irreversible Isothermal Expansion
When expansion occurs against a constant external pressure ($P_{ext}$) quickly (e.g., free expansion or single step).
Work Done (Irreversible)
Free Expansion (vacuum): $P_{ext} = 0 \Rightarrow w = 0$.
Knowledge Check
Test your understanding of Isothermal Processes
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