An ideal gas expands adiabatically from an initial temperature `T_(1)` to a final temperature `T_(2)`. Prove that the work done by the gas is `C_(V) (T_(1) - T_(2))`.

Explanation

To prove that the work done by an ideal gas during an adiabatic expansion from an initial temperature \( T_1 \) to a final temperature \( T_2 \) is given by \( W = C_V (T_1 - T_2) \), we will follow these steps:

### Step 1: Understand the Adiabatic Process
In an adiabatic process, there is no heat exchange with the surroundings. Therefore, the heat transfer \( dQ = 0 \).

### Step 2: Apply the First Law of Thermodynamics
The first law of thermodynamics states:
\[
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