1. **State the problem:** We want to calculate the pressure $P$ of a real gas using the van der Waals equation of state:
$$\left(P + \frac{n^2 a}{V^2}\right)(V - nb) = nRT$$
Given:
- Temperature $T = 573$ K
- Volume $V = 1$ L = $1 \times 10^{-3}$ m$^3$ (convert liters to cubic meters)
- Moles $n = 2$ mol
- Constants $a = 5.536$ L$^2 \cdot$ bar / mol$^2$, $b = 3.049 \times 10^{-2}$ m$^3$/mol
- Gas constant $R = 8.314$ kPa$\cdot$m$^3$/kmol$\cdot$K
2. **Convert units for consistency:**
- Convert $a$ from L$^2 \cdot$ bar / mol$^2$ to m$^6 \cdot$ kPa / mol$^2$:
- $1$ L = $10^{-3}$ m$^3$ so $1$ L$^2 = (10^{-3})^2 = 10^{-6}$ m$^6$
- $1$ bar = $100$ kPa
- So, $a = 5.536 \times 10^{-6} \times 100 = 5.536 \times 10^{-4}$ m$^6 \cdot$ kPa / mol$^2$
- Convert $V$ from L to m$^3$: $V = 1$ L = $1 \times 10^{-3}$ m$^3$
- Convert $n$ from mol to kmol: $n = 2$ mol = $0.002$ kmol
3. **Rewrite the van der Waals equation to solve for $P$:**
$$P = \frac{nRT}{V - nb} - \frac{n^2 a}{V^2}$$
4. **Calculate each term:**
- Calculate $nb = n \times b = 0.002 \times 3.049 \times 10^{-2} = 6.098 \times 10^{-5}$ m$^3$
- Calculate $V - nb = 1 \times 10^{-3} - 6.098 \times 10^{-5} = 9.3902 \times 10^{-4}$ m$^3$
- Calculate $nRT = 0.002 \times 8.314 \times 573 = 9.528$ kPa$\cdot$m$^3$
- Calculate first term $\frac{nRT}{V - nb} = \frac{9.528}{9.3902 \times 10^{-4}} = 10153.5$ kPa
- Calculate second term $\frac{n^2 a}{V^2} = \frac{(0.002)^2 \times 5.536 \times 10^{-4}}{(1 \times 10^{-3})^2} = \frac{4 \times 10^{-6} \times 5.536 \times 10^{-4}}{1 \times 10^{-6}} = 2.2144$ kPa
5. **Calculate pressure $P$:**
$$P = 10153.5 - 2.2144 = 10151.3 \text{ kPa}$$
6. **Calculate ideal gas pressure $P_{IGL}$:**
$$P_{IGL} = \frac{nRT}{V} = \frac{9.528}{1 \times 10^{-3}} = 9528 \text{ kPa}$$
7. **Calculate percent difference:**
$$100 \times \left| \frac{P_{vdw} - P_{IGL}}{P_{vdw}} \right| = 100 \times \left| \frac{10151.3 - 9528}{10151.3} \right| = 6.17\%$$
---
**MATLAB code:**
```matlab
% Given values
T = 573; % K
V_L = 1; % L
n_mol = 2; % mol
a_L2bar_per_mol2 = 5.536; % L^2*bar/mol^2
b_m3_per_mol = 3.049e-2; % m^3/mol
R_kPa_m3_per_kmol_K = 8.314; % kPa*m^3/kmol*K
% Unit conversions
V = V_L * 1e-3; % L to m^3
n = n_mol * 1e-3; % mol to kmol
a = a_L2bar_per_mol2 * 1e-6 * 100; % L^2*bar/mol^2 to m^6*kPa/mol^2
% Calculate pressure using van der Waals equation
P_vdw = (n*R_kPa_m3_per_kmol_K*T) / (V - n*b_m3_per_mol) - (n^2 * a) / V^2;
% Calculate ideal gas pressure
P_IGL = (n*R_kPa_m3_per_kmol_K*T) / V;
% Calculate percent difference
percent_diff = 100 * abs((P_vdw - P_IGL) / P_vdw);
% Display results
fprintf('P_vdw = %.2f kPa\n', P_vdw);
fprintf('P_IGL = %.2f kPa\n', P_IGL);
fprintf('Percent difference = %.2f%%\n', percent_diff);
```
This code calculates the van der Waals pressure, ideal gas pressure, and their percent difference for the given inputs.
Van Der Waals E03392
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