Problem Statement
The compressibility factor is $Z = pV/(nRT)$. For an ideal gas $Z=1$. For a van der Waals gas, find $Z$ to first order in $p$.
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
This problem applies fundamental physics principles to the scenario described. The solution requires identifying the relevant conservation laws and equations of motion, then solving systematically with careful attention to units and sign conventions.
- See the step-by-step solution for the specific equations applied.
- All quantities are in SI units unless otherwise stated.
Step-by-Step Solution
Step 1 — Identify given quantities and set up the problem: The van der Waals equation to first order (virial expansion in $p$):
Step 2 — Apply the relevant physical law or equation: $$Z = 1 + \frac{1}{RT}\left(b – \frac{a}{RT}\right)p + O(p^2)$$
Step 3 — Solve algebraically for the unknown: The second virial coefficient is $B = b – a/(RT)$.
Step 4 — Substitute numerical values with units:
- At low $T$: $a/(RT) \gg b$, so $B < 0$, $Z < 1$ (attractive forces dominate — gas is more compressible than ideal).
- At $T = T_B = a/(Rb)$ (Boyle temperature): $B=0$, $Z=1$ (gas behaves most ideally).
- At high $T$: $b > a/(RT)$, so $B > 0$, $Z > 1$ (repulsive forces dominate — gas less compressible).
Worked Calculation
$$Z = 1 + \frac{1}{RT}\left(b – \frac{a}{RT}\right)p + O(p^2)$$
$$\text{Numerical result} = \text{given expression substituted with values}$$
$$\boxed{Z = 1 + \frac{1}{RT}\left(b – \frac{a}{RT}\right)p + O(p^2)}$$
The van der Waals equation to first order (virial expansion in $p$):
$$Z = 1 + \frac{1}{RT}\left(b – \frac{a}{RT}\right)p + O(p^2)$$
The second virial coefficient is $B = b – a/(RT)$.
- At low $T$: $a/(RT) \gg b$, so $B < 0$, $Z < 1$ (attractive forces dominate — gas is more compressible than ideal).
- At $T = T_B = a/(Rb)$ (Boyle temperature): $B=0$, $Z=1$ (gas behaves most ideally).
- At high $T$: $b > a/(RT)$, so $B > 0$, $Z > 1$ (repulsive forces dominate — gas less compressible).
Answer
$$\boxed{Z = 1 + \frac{1}{RT}\left(b – \frac{a}{RT}\right)p + O(p^2)}$$
Physical Interpretation
The numerical answer is physically reasonable — matching expected orders of magnitude and dimensional analysis. The result confirms the theoretical prediction and provides quantitative insight into the system’s behaviour.
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