Problem 116 - Chemistry The Central Science In Si Units

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.)

Step-by-Step Solution

Answer

At STP, Ar atoms occupy about 0.14% of the volume; at 20.27 MPa and 0°C, they occupy about 7.23%.

Step 1: Understanding the Problem

We are given that the van der Waals constant, \(b\), is four times the volume occupied by a mole of gas molecules. We need to calculate the fraction of volume occupied by Ar atoms at two different conditions: standard temperature and pressure (STP) and high pressure with a different temperature.

Step 2: Calculate Molar Volume at STP

At STP, by definition, 1 mole of an ideal gas occupies 22.414 liters. This is our total volume \(V_{STP}\) for 1 mole.

Step 3: Calculate Volume Occupied by Ar Molecules at STP

For Ar, we assume the van der Waals constant \(b\) is known. Using \(b = 4V_{molecules}\), the volume occupied by Ar molecules \(V_{molecules} = \frac{b}{4}\). We use the known value of \(b = 0.0322 \, ext{L/mol} \) for Argon.

Step 4: Calculate Fraction of Volume Occupied at STP

The fraction of volume that Ar atoms occupy at STP is given by \( \frac{V_{molecules}}{V_{STP}} = \frac{0.0322}{22.414} \approx 0.001437\). This is the fraction of the container volume occupied by Ar atoms at STP.

Step 5: Calculate Molar Volume at High Pressure

Using the ideal gas law \(PV = nRT\), where \(P = 20.27 \, ext{MPa} = 20270 \, ext{kPa}\), \(n = 1\), \(R = 8.314 \, ext{J/(mol} \, \text{K})\), and \(T = 273.15 \, ext{K}\), solve for \(V\): \[ V = \frac{nRT}{P} = \frac{1 \times 8.314 \times 273.15}{20270} \approx 0.1114 \, ext{L}\]. This is the volume \(V_{high}\) at the given conditions.

Step 6: Calculate Fraction of Volume Occupied at High Pressure

Again, using \(V_{molecules} = \frac{b}{4} = \frac{0.0322}{4} \), the fraction of volume occupied by Ar atoms at high pressure is \( \frac{0.00805}{0.1114} \approx 0.0723\). This is the fraction of the container volume occupied by Ar atoms at the specified high pressure and \(0^{\circ}C\).

Key Concepts

Ideal Gas Law

The ideal gas law is a fundamental principle in chemistry and physics that describes the behavior of gases under various conditions. It is often represented by the equation \( PV = nRT \), where:

• \( P \) stands for pressure,
• \( V \) represents volume,
• \( n \) is the amount of substance in moles,
• \( R \) is the ideal gas constant (8.314 J/(mol·K)), and
• \( T \) stands for temperature in Kelvin.

This equation helps us understand how a gas will behave when subjected to changes in these variables.
The ideal gas law assumes that gas molecules do not interact and occupy no volume. Though this is not true for real gases, it is an excellent approximation under normal conditions. For example, it allows us to calculate the volume of a gas given its pressure and temperature, as demonstrated when we determined the molar volume of argon under high-pressure conditions.

Molar Volume

Molar volume refers to the volume occupied by one mole of a substance. For gases, it is often discussed at standard temperature and pressure (STP).
At STP, one mole of an ideal gas occupies a volume of 22.414 liters. This is a key reference point for understanding gas behavior.

• Using the ideal gas law, you can derive this value: when \( P = 101.3 \, \text{kPa} \) and \( T = 273.15 \, \text{K} \), the volume \( V \) for one mole of gas can be calculated, resulting in 22.414 liters.
• Even when using the simplistic assumptions provided by the ideal gas equation, the molar volume helps in predicting and understanding the observed behaviors under varying conditions.

Thus, knowing the molar volume is particularly essential in determining how much space a given amount of gas will occupy at ideal conditions.

Standard Temperature and Pressure (STP)

Standard Temperature and Pressure (STP) are specific values of temperature and pressure used as standard reference conditions in scientific calculations.
STP is defined as a temperature of 0°C (273.15 K) and a pressure of 1 atm (101.3 kPa). These conditions are used to standardize measurements and calculations involving gases, allowing scientists to compare results more easily across different experiments.

• At STP, an ideal gas occupies a molar volume of approximately 22.414 liters per mole.
• The concept of STP is crucial for calculations that involve gas properties like density, volume, and pressure.
• In the context of the problem, STP serves as the benchmark to calculate how much of the container's volume is actually occupied by argon atoms, providing a clear picture of gas behavior in an often complex system.

Overall, STP is an essential standard for scientists and engineers when discussing or calculating the properties of gases under consistent and controlled conditions.