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Chapter 10: Problem 116

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. Using this figure, calculate the fraction of the volume in a container actually occupied by Ar atoms (a) at STP, (b) at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\). (Assume for simplicity that the ideal-gas equation still holds.)

Short Answer

Expert verified
The fraction of the volume in a container actually occupied by Argon atoms is 0.25 or 25% at both (a) STP (Standard Temperature and Pressure) and (b) at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\).

Step by step solution

01

Recall the ideal gas law and the relationship between van der Waals constant b and volume occupied by a mole of gas

The ideal gas law is given by the equation: \(PV = nRT\), where P is the pressure, V is the volume, n is the number of moles, R is the gas constant, and T is the temperature in Kelvin. For Argon, the van der Waals constant b is given as \(b=4 V_m\), where \(V_m\) is the volume occupied by a mole of Argon atoms.
02

Calculate the molar volume at STP

At STP (Standard Temperature and Pressure), the pressure is 100 kPa, and the temperature is \(273.15 K\). We can use the ideal gas law to calculate the molar volume of Argon at STP. Using the ideal gas law equation: \(PV = nRT\) \(V_m = \frac{nRT}{P} = \frac{R \times 273.15}{100 \times 10^3}\) Note: At STP, we assume one mole of any gas, so n=1. Also, the gas constant R is equal to \(8.314 J/(mol \cdot K)\).
03

Calculate the volume occupied by Argon at STP

Substitute the values for R, T, and P from the previous step: \(V_m = \frac{8.314 \times 273.15}{100 \times 10^3} = 0.0224\,m^3/mol\) Now, we can use the relationship between the van der Waals constant b and the volume occupied by a mole of gas to calculate the volume occupied by one mole of Argon: \(b = 4V_m \Rightarrow V_{occupied} = \frac{b}{4} = \frac{0.0224}{4} = 0.0056\,m^3/mol\)
04

Calculate the fraction of the volume occupied by Argon at STP

To find the fraction of volume occupied by Argon atoms, divide the volume occupied by Argon by the molar volume: Fraction of volume occupied = \(\frac{V_{occupied}}{V_m} = \frac{0.0056}{0.0224} = 0.25\) So, the fraction of the volume in a container actually occupied by Argon atoms at STP is 0.25 or 25%.
05

Calculate the molar volume at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\)

Now, we will calculate the molar volume of Argon at a pressure of 20.27 MPa and a temperature of \(0^{\circ} \mathrm{C}\) (273.15 K). Using the ideal gas law equation: \(PV = nRT\) \(V_m = \frac{nRT}{P} = \frac{R \times 273.15}{20.27 \times 10^6}\)
06

Calculate the volume occupied by Argon at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\)

Substitute the values for R, T, and P from the previous step: \(V_m = \frac{8.314 \times 273.15}{20.27 \times 10^6} = 1.11 \times 10^{-4} m^3/mol\) Now, we can use the relationship between the van der Waals constant b and the volume occupied by a mole of gas to calculate the volume occupied by one mole of Argon: \(V_{occupied} = \frac{b}{4} = \frac{1.11 \times 10^{-4}}{4} = 2.78 \times 10^{-5} m^3/mol\)
07

Calculate the fraction of the volume occupied by Argon at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\)

To find the fraction of volume occupied by Argon atoms, divide the volume occupied by Argon by the molar volume: Fraction of volume occupied = \(\frac{V_{occupied}}{V_m} = \frac{2.78 \times 10^{-5}}{1.11 \times 10^{-4}} = 0.25\) So, the fraction of the volume in a container actually occupied by Argon atoms at 20.27 MPa pressure and \(0^{\circ} \mathrm{C}\) is 0.25 or 25%.

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Most popular questions from this chapter

A sample of \(5.00 \mathrm{~mL}\) of diethylether $\left(\mathrm{C}_{2} \mathrm{H}_{5} \mathrm{OC}_{2} \mathrm{H}_{5},\right.\( density \)=0.7134 \mathrm{~g} / \mathrm{mL}\( ) is introduced into a \)6.00-\mathrm{L}$ vessel that already contains a mixture of \(\mathrm{N}_{2}\) and \(\mathrm{O}_{2}\), whose partial pressures are \(R_{\mathrm{N}_{2}}=21.08 \mathrm{kPa}\) and \(P_{\mathrm{O}_{2}}=76.1 \mathrm{kPa}\). The temperature is held at \(35.0^{\circ} \mathrm{C},\) and the diethylether totally evaporates. (a) Calculate the partial pressure of the diethylether. (b) Calculate the total pressure in the container.

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