It turns out that the van der Waals constant b
equals four times the total volume actually occupied by the molecules of a mole of gas. Calculate the fraction of the volume in a container actually occupied by Ar
atoms (b
= 0.0322 L/mol
):
at 150 atm
pressure and 0 ∘C
. (Assume for simplicity that the ideal-gas equation still holds.)
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To calculate the fraction of the volume in a container actually occupied by Ar atoms, we can use the van der Waals equation:
\[ \left(P + \frac{a}{V^2}\right)(V-b) = RT \]
Given:
Pressure (P) = 150 atm
Temperature (T) = 0 °C = 273 K
van der Waals constant b = 0.0322 L/mol
We also know that b equals four times the total volume actually occupied by the molecules of a mole of gas:
\[ b = 4 \times \text{total volume actually occupied by the gas molecules} \]
Since one mole of Ar occupies 22.4 L at STP, it occupies 22.4/24.45 = 0.918 L at 0°C Temperature and 1 atm pressure.
Therefore, the total volume actually occupied by Ar molecules in the given conditions is 4 times this value:
\[ \text{Total volume} = 4 \times 0.918 = 3.67 L \]
Now, we need to solve the van der Waals equation for the actual volume occupied by Ar molecules:
\[ (150 + \frac{a}{V^2})(V - 0.0322) = 0.0821 \times 273 \]
Solving this equation will give us the actual volume occupied by Ar molecules in the given conditions. Once we have this volume, we can calculate the fraction of the volume in the container actually occupied by Ar atoms.