Question

Write the mathematical equation of first law of thermodynamics for following processes : (a) Isochoric process, (b) Adiabatic process

Hint:

Isochoric $\rightarrow$ constant volume $\rightarrow$ no work.
Adiabatic $\rightarrow$ insulated system $\rightarrow$ no heat.

Answer 1

Step 1: Understanding the Concept:
The First Law of Thermodynamics is stated as: \( \Delta U = q + w \).
Different thermodynamic processes impose constraints on \( q \) or \( w \).
Step 2: Detailed Explanation:
(a) Isochoric process:
In an isochoric process, the volume remains constant (\( \Delta V = 0 \)).
Work done is given by \( w = -P_{ext} \Delta V \). Since \( \Delta V = 0 \), \( w = 0 \).
Substituting in the first law:
\[ \Delta U = q + 0 \Rightarrow \Delta U = q_v \]
This means the change in internal energy is equal to the heat exchanged at constant volume.

(b) Adiabatic process:
In an adiabatic process, there is no exchange of heat between the system and surroundings (\( q = 0 \)).
Substituting in the first law:
\[ \Delta U = 0 + w \Rightarrow \Delta U = w \]
This means the change in internal energy is equal to the adiabatic work done.
Step 3: Final Answer:
For Isochoric: \( \Delta U = q_v \). For Adiabatic: \( \Delta U = w \).