A $1.00 L$ vessel contains nitrogen ( $a = 1.39 L^{2} atm mo l^{- 2}$ , $b = 0.0391 L mo l^{- 1}$ ) at $300 K$ and $30.0 atm$ . Solve the van der Waals equation for the amount of gas by iteration, starting from the ideal value and using $n = \frac{( P + a n ^{2} / V ^{2} ) ( V - nb )}{RT}$ . Give $n$ in mol to three decimals ( $R = 0.0821 L atm K^{- 1} mo l^{- 1}$ ).

Numerical answer type, answer in mol, tolerance 0.003.

Answer: 1.242 mol

Ideal:

n = 30.0/24.63 = 1.218

. Iteration 1:

a n^{2} / V^{2} = 2.062

V - nb = 0.9524

n = 32.062 \times 0.9524/24.63 = 1.240

. Iteration 2:

a n^{2} / V^{2} = 2.137

V - nb = 0.9515

n = 32.137 \times 0.9515/24.63 = 1.242

. Converged:

n = 1.2417 mol

, i.e.

1.242 mol

, about 2 percent more than the ideal estimate.