Question

Table 10.3 shows that the van der Waals \(b\) parameter has units of \(\mathrm{L} / \mathrm{mol}\). This implies that we can calculate the size of atoms or molecules from \(b\). Using the value of \(b\) for \(\mathrm{Xe},\) calculate the radius of a Xe atom and compare it to the value found in Figure \(7.7,\) that is, \(140 \mathrm{pm}\). Recall that the volume of a sphere is \((4 / 3) \pi r^{3}\).

Short answer

To find the radius of a Xe atom using its van der Waals \(b\) parameter, first find the volume of a single Xe atom by dividing the \(b\) parameter value (in cm³/mol) by Avogadro's number (\(6.022\times10^{23}\) atoms/mol). Then, use the volume of a sphere formula, \(V=\frac{4}{3}\pi r^3\), and solve for the radius: \(r = \sqrt[3]{\frac{3\cdot V_{\text{Xe-atom}}}{4\cdot\pi}}\) Calculate the radius in picometers (1 cm = \(10^8\) pm) and compare it to the given value of 140 pm.

Step-by-step solution

Calculate the volume of one Xe atom

Since the \(b\) parameter of the van der Waals equation represents the volume occupied by one mole of the substance, we need to calculate the volume of one Xe atom. For that, we will first convert L/mol into the appropriate units and then divide the b parameter value by Avogadro's number (\(N_{A}=6.022\times10^{23}\) atoms/mol).

Convert the given \(b\) parameter value into appropriate units

In this step, we are going to convert the value of the \(b\) parameter given in L/mol into a smaller unit. It's more convenient to work with the volume of atoms in cubic centimeters, so to convert 1 L = 1000 cm³

Find the \(b\) parameter value of Xe

According to the problem statement, the \(b\) parameter value for Xe should be found in Table 10.3. Look up the value in the table and denote it as \(b_X\).

Calculate the volume of a single Xe atom

Now, use Avogadro's number and the \(b\) parameter to find the volume of one Xe atom as follows: \(V_{\text{Xe-atom}} = \frac{b_X}{N_A}\) Make sure to use the \(b_X\) value converted to cm³/mol in Step 2.

Calculate the radius of the Xe atom

Using the volume of a single Xe atom calculated in the previous step, we will now find its radius. Since the Xe atom can be approximated as a sphere, we can use the formula for the volume of a sphere as \(V=\frac{4}{3}\pi r^3\), and write the formula to find the radius \(r\): \(r = \sqrt[3]{\frac{3\cdot V_{\text{Xe-atom}}}{4\cdot\pi}}\) Calculate the radius of the Xe atom using the formula above.

Compare the calculated radius with the given value

The problem states that the value of the radius found in Figure 7.7 is 140 pm (picometers). Convert the calculated radius from centimeters to picometers: 1 cm = \(10^8\) pm Compare the calculated radius with the given value of 140 pm and analyze the discrepancy.

Key concepts

Atomic Radius

The atomic radius is a measure of the size of an atom. It is typically defined as the distance from the center of the nucleus to the outer boundary of its electron cloud. Understanding and calculating atomic radius is crucial in chemistry because it helps in predicting and explaining the properties of elements and how they interact with each other.

In the van der Waals equation, the volume occupied by individual atoms or molecules affects how gases behave under different conditions. By utilizing the van der Waals parameter \(b\), one can indirectly determine the atomic radius. For instance, in the exercise, the radius of a Xenon (Xe) atom is calculated from its \(b\) parameter. By knowing the volume of the atom, derived from its spherical nature \(V = \frac{4}{3} \pi r^3\), the radius can be determined as a significant factor.

Avogadro's Number

Avogadro's number is one of the fundamental constants in chemistry, indicating the number of atoms, molecules, or particles in one mole of a substance. This number is approximately \(6.022 \times 10^{23}\) particles/mol.

In our context, Avogadro's number plays a pivotal role in translating macroscopic measurements into microscopic observations. When we use the van der Waals equation, the \(b\) parameter provides the volume occupied by a mole of atoms or molecules. By dividing this volume by Avogadro's number, we get the volume occupied by a single atom or molecule, which is necessary to calculate the atomic radius. This process highlights how Avogadro's number allows for the conversion between the scale of atomic particles and measureable quantities in the laboratory.

Atomic Volume

The atomic volume refers to the volume occupied by an atom of an element. It is an essential parameter in understanding material properties and calculating atomic radii. When discussing atomic volumes, we often rely on the translations of macroscopic volume measurements to atomic scales.

The van der Waals \(b\) parameter is directly related to atomic volume. Typically, it represents the volume taken up by one mole of a gas. To determine the volume of a single atom from the \(b\) parameter, the value is divided by Avogadro's number. The result is crucial for determining properties such as the atomic radius and helps in comprehending how atoms interact within substances.

van der Waals Parameters

The van der Waals parameters are a set of adjustments introduced to account for deviations of real gases from ideal gas behavior. These parameters, identified as \(a\) and \(b\), modify the ideal gas law to better reflect the interactions and volumes of gas particles.

The parameter \(b\) is particularly useful for calculating the volume of atoms or molecules. It accounts for the finite size of these particles, which is an important factor absent from the ideal gas law. In the exercise, the \(b\) parameter provides a means to calculate the atom's volume, leading to the determination of atomic radius. Understanding these parameters is key to mastering gas-related problems and accurately predicting gas behavior under varied experimental conditions.