Question

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

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

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.

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 \(8.314 J/(mol \cdot K)\).

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: \(b = 4V_m \Rightarrow V_{occupied} = \frac{b}{4} = \frac{0.0224}{4} = 0.0056\,m^3/mol\)

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%.

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: \(PV = nRT\) \(V_m = \frac{nRT}{P} = \frac{R \times 273.15}{20.27 \times 10^6}\)

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: \(V_{occupied} = \frac{b}{4} = \frac{1.11 \times 10^{-4}}{4} = 2.78 \times 10^{-5} m^3/mol\)

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%.

Key concepts

Ideal Gas Law

The ideal gas law is a cornerstone concept in the study of gases. It provides a simple relationship between pressure, volume, temperature, and the amount of gas. The equation for the ideal gas law is:\( PV = nRT \)- **P:** Pressure of the gas in pascals (Pa) or kilopascals (kPa).- **V:** Volume of the gas in cubic meters (m³).- **n:** Number of moles of gas.- **R:** Universal gas constant, which is approximately \(8.314 \, J/(mol \cdot K)\).- **T:** Temperature in Kelvin (K).At standard temperature and pressure (STP), which is defined as a temperature of 273.15 Kelvin and a pressure of 100 kPa, the ideal gas law can calculate the volume a mole of gas will occupy. For all practical purposes, at STP, one mole of any gas occupies a volume of approximately 22.4 liters or 0.0224 m³. This law works remarkably well under conditions where intermolecular forces are negligible and the gas does not occupy much volume.

Molar Volume

Molar volume is an important concept when dealing with gases and it refers to the volume occupied by one mole of gas. At STP, as mentioned, the molar volume of an ideal gas is about 22.4 liters per mole.The formula for molar volume using the ideal gas law is:\( V_m = \frac{nRT}{P} \)Since \(n = 1\) mole for molar calculations, it simplifies calculations by directly relating gas constants and conditions (temperature and pressure) to find the volume occupied by a mole of gas.- At STP, substitute the standard temperature (273.15 K) and pressure (100 kPa) into the ideal gas law and it confirms the molar volume: \( V_m = \frac{8.314 \times 273.15}{100 \times 10^3} = 0.0224\, m^3 /mol\).Understanding molar volume helps predict how much space a given amount of gas will occupy under certain conditions and is crucial in chemistry and physics where gas behavior modeling is essential.

Fraction of Volume Occupied

The fraction of volume occupied by gas molecules within a container is significant in understanding gas behavior beyond ideal conditions. Though gases appear to occupy their entire container, much of this space is empty.To determine the actual fraction occupied by gas particles:- Calculate the volume truly occupied by the molecules. This can be linked to the van der Waals constant, \(b\), which gives the measure of the volume that gas molecules occupy due to their size.- Using the relationship given in this exercise, \( b = 4V_{occupied} \), the occupied volume \(V_{occupied}\) for one mole is: \( V_{occupied} = \frac{b}{4} \).- Compare this occupied volume with the molar volume, \(V_m\), from the ideal gas law.- The fraction of volume occupied is: \( \text{Fraction occupied} = \frac{V_{occupied}}{V_m} \).This fraction tells us how closely packed the gas particles are and provides insight into the behavior of real gases versus ideal gases. At both standard and high pressures, this fraction remains fairly constant at around 25% for Argon due to molecular volume considerations, indicating significant empty space in real gases.