A sample of an ideal gas is expanded to twice its original volume of `1m^(3)` in a reversible process for which `P = alphaV^(2)`, where `alpha = 5 atm m^(-6)`. If `C_(V,m) = 20 J mol^(-1)K^(-1)`, determine moalr change in entropy `(DeltaS_(m))` for the process.

Explanation

To determine the molar change in entropy \((\Delta S_m)\) for the process where a sample of an ideal gas is expanded to twice its original volume in a reversible process, we can follow these steps:

### Step 1: Understand the Process
We know that the gas is expanded from an initial volume \(V_1 = 1 \, m^3\) to a final volume \(V_2 = 2 \, m^3\). The pressure is given by the equation \(P = \alpha V^2\), where \(\alpha = 5 \, \text{atm} \, m^{-6}\).

### Step 2: Use the First Law of Thermodynamics
According to the first law of thermodynamics:
\[
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