The compressibility factor (Z) is defined as:
\( Z = \frac{PV}{RT} \)
For a van der Waals gas, the equation of state is:
\( \left(P + \frac{a}{V^2}\right)(V - b) = RT \)
At high pressures, the volume V decreases, making the term b (excluded volume) significant. The term involving a (intermolecular attraction) becomes negligible. Simplifying for high pressure:
\( Z \approx 1 + \frac{Pb}{RT} \)
Thus, the compressibility factor is approximately \( 1 + \frac{Pb}{RT} \) under high-pressure conditions.
Correct Answer:
Option 2: 1 + Pb/RT
Explanation:
Van der Waals Equation:
(P + a(n/V)2) (V - nb) = nRT
Compressibility Factor (Z):
Z = PV / nRT
High-Pressure Approximation:
At high pressure, V is small, and 'nb' is significant. 'a(n/V)2' becomes negligible compared to P.
Therefore, P(V - nb) ≈ nRT
PV - Pnb ≈ nRT
PV ≈ nRT + Pnb
PV / nRT ≈ 1 + Pnb / nRT
Z ≈ 1 + Pb / RT
\( \text{M} \xrightarrow{\text{CH}_3\text{MgBr}} \text{N} + \text{CH}_4 \uparrow \xrightarrow{\text{H}^+} \text{CH}_3\text{COCH}_2\text{COCH}_3 \)
Identify the ion having 4f\(^6\) electronic configuration.