Question

the van der waals equation of state was designed (by dutch physicist johannes van der waals) to predict the relationship between pressure p, volume v and temperature t for gases better than the ideal gas law does:

$$ \\left( p + a \\frac { n ^ { 2 } } { v ^ { 2 } } \ ight) ( v - n b ) = n r t $$

the van der waals equation of state. r stands for the gas constant and n for moles of gas.

the parameters a and b must be determined for each gas from experimental data.

use the van der waals equation to answer the questions in the table below.

Explanation:

Step1: Convert temperature to Kelvin

$T = 165.0+273.15=438.15\ K$

Step2: Calculate pressure using van der Waals equation

Given $a = 4.169\ \frac{L^{2}\cdot atm}{mol^{2}}$, $b = 0.0371\ \frac{L}{mol}$, $V = 0.445\ L/mol$, $R=0.0821\ \frac{L\cdot atm}{mol\cdot K}$
First, calculate $\frac{a}{V^{2}}$:
$\frac{a}{V^{2}}=\frac{4.169}{(0.445)^{2}}=\frac{4.169}{0.198025}\approx21.05$
Then, calculate $V - b$:
$V - b=0.445 - 0.0371 = 0.4079$
Now, from $(p+\frac{a}{V^{2}})(V - b)=RT$
$p+\frac{a}{V^{2}}=\frac{RT}{V - b}$
$p=\frac{RT}{V - b}-\frac{a}{V^{2}}$
Substitute the values:
$\frac{RT}{V - b}=\frac{0.0821\times438.15}{0.4079}=\frac{35.971115}{0.4079}\approx88.19$
$p = 88.19-21.05=67.14\approx67.1\ atm$

Step3: Calculate pressure using Ideal Gas Law

From $pV = RT$
$p=\frac{RT}{V}$
Substitute $R = 0.0821\ \frac{L\cdot atm}{mol\cdot K}$, $T = 438.15\ K$, $V = 0.445\ L/mol$
$p=\frac{0.0821\times438.15}{0.445}=\frac{35.971115}{0.445}\approx80.8\ atm$

Answer:

• Van der Waals pressure: $67.1\ atm$
• Ideal Gas Law pressure: $80.8\ atm$