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Chapter 10: Problem 95

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

Expert verified
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

01

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).
02

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
03

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 \(b_X\).
04

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.
05

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.
06

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.

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