Problem Statement
Solve the fluid mechanics problem: Find the pressure of $\nu = 1.0\ \text{mol}$ of CO₂ in $V = 0.50\ \text{L}$ at $T = 300\ \text{K}$ using the van der Waals equation. ($a = 0.364\ \text{J·m}^3/\text{mol}^2$, $b = 42.9\ \text{cm}^3/\text{mol}$) $$p = \frac{\nu RT}{V-\nu b} – \frac{a\nu^2}{V^2}$$ With $V = 5.0\times10^{-4}\ \text{m}^3
Given Information
- See problem statement for all given quantities.
Physical Concepts & Formulas
The van der Waals equation of state corrects the ideal gas law for finite molecular volume and intermolecular attractions. The parameter $a$ accounts for molecular attraction (reducing effective pressure) and $b$ for excluded volume (reducing available volume).
- $(P + a/V_m^2)(V_m – b) = RT$ — van der Waals equation (per mole)
- Critical point: $T_c = 8a/(27Rb)$, $P_c = a/(27b^2)$, $V_{m,c} = 3b$
- Compressibility factor: $Z = PV_m/RT$
Step-by-Step Solution
Step 1 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 2 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Step 3 — Verify the result: Check units, limiting cases, and order of magnitude to confirm the answer is physically reasonable.
Worked Calculation
$$p = \frac{\nu RT}{V-\nu b} – \frac{a\nu^2}{V^2}$$
Answer
$$\boxed{v_{\text{efflux}} = \sqrt{2gh}}$$
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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