Question

The van der Waal's equation \(\left(P + \frac{a}{4V^2}\right)\left(V - \frac{b}{2}\right) = \frac{RT}{2}\) is valid for:

• A. 1 mole of an ideal gas.
• B. 2 moles of a real gas.
• C. $\frac{1}{2}$ mole of an ideal gas.
• D. $\frac{1}{2}$ mole of a real gas.

Hint:

To quickly solve such problems, always compare the coefficients of the constant terms on the right-hand side ($nRT$) and the volume correction term ($nb$). Here, comparing $nRT = \frac{RT}{2}$ immediately gives $n = \frac{1}{2}$ without any tedious calculations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The van der Waals equation is the equation of state for real gases. It modifies the ideal gas law (\(PV=nRT\)) by introducing corrections for intermolecular forces (\(a\)) and the finite volume of gas molecules (\(b\)).
The standard form of the equation for \(n\) moles is:
\[ \left( P + \frac{n^2 a}{V^2} \right) (V - nb) = nRT \]
Step 2: Key Formula or Approach:
To solve this, we compare the coefficients of the given specific equation with the standard general equation to determine the value of \(n\).
Step 3: Detailed Explanation:
The given equation is:
\[ \left( P + \frac{a}{4V^2} \right) \left( V - \frac{b}{2} \right) = \frac{RT}{2} \]
Let's compare the corresponding terms:
1. Pressure Correction Term:
Standard: \(n^2 a / V^2\). Given: \(a / (4V^2)\).
Equating them: \(n^2 = 1/4 \Rightarrow n = 1/2\).
2. Volume Correction Term:
Standard: \(nb\). Given: \(b/2\).
Equating them: \(n = 1/2\).
3. Ideal Term (Right Hand Side):
Standard: \(nRT\). Given: \(RT/2\).
Equating them: \(n = 1/2\).
All three terms consistently yield \(n = 1/2\).
Since the van der Waals equation (containing \(a\) and \(b\)) only applies to real gases (ideal gases have \(a=0\) and \(b=0\)), the equation is valid for \(1/2\) mole of a real gas.
Step 4: Final Answer:
The equation is valid for \(1/2\) mole of a real gas. The correct option is (D).