Find Ratio of Pressure to Ideal Molar Volume for Real Gas

To find the value of $\mathbf{x}/\mathbf{y}$, we use the definition of the compressibility factor ($Z$) and the ideal gas equation.

Step 1: Calculate the pressure $x$ using the compressibility factor

The compressibility factor $Z$ of a real gas is defined as: $Z = \frac{P V_m}{R T}$

Given:

• $Z = 0.5$
• $V_m = 0.4 \text{ dm}^3\text{ mol}^{-1} = 0.4 \text{ L mol}^{-1}$
• $T = 800 \text{ K}$
• $P = x \text{ atm}$
• $R = 8 \times 10^{-2} \text{ L atm K}^{-1}\text{ mol}^{-1} = 0.08 \text{ L atm K}^{-1}\text{ mol}^{-1}$

Substitute the given values into the formula: $0.5 = \frac{x \times 0.4}{0.08 \times 800}$

$0.5 = \frac{0.4 x}{64}$

Solve for $x$: $0.4 x = 0.5 \times 64 = 32$ $x = \frac{32}{0.4} = 80 \text{ atm}$

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Step 2: Calculate the ideal molar volume $y$

For an ideal gas at the same temperature ($T = 800 \text{ K}$) and pressure ($P = x = 80 \text{ atm}$): $P y = R T$

Alternatively, since $Z = \frac{V_m}{V_{m,\text{ideal}}} = \frac{V_m}{y}$, we have: $0.5 = \frac{0.4}{y}$ $y = \frac{0.4}{0.5} = 0.8 \text{ dm}^3\text{ mol}^{-1}$

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Step 3: Calculate the value of $x/y$

$\frac{x}{y} = \frac{80}{0.8} = 100$