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Chapter 10: Problem 99

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 Xe, calculate the radius of a Xe atom and compare it to the value found in Figure 7.6, \(1.30 \AA\) A. Recall that the volume of a sphere is \((4 / 3) \pi r^{3}\).

Short Answer

Expert verified
The radius of a Xe atom can be calculated using the formula \(r = \sqrt[3]{\frac{3 b}{4 \pi N_A}}\), where \(b\) is the van der Waals \(b\) parameter for Xe and \(N_A\) is Avogadro's number. Comparing the calculated radius with the given value of \(1.30 \unicode{x212B}\) A confirms that the van der Waals \(b\) parameter can be used to determine the size of atoms or molecules.

Step by step solution

01

Write the expression for the b parameter in terms of the volume of a sphere

Recall that the volume of a sphere can be calculated using the formula: \(V = \frac{4}{3} \pi r^3\). Also, it is given that the van der Waals b parameter has units of L/mol, implying that the b parameter can be related to the volume of atoms or molecules per mole. In this case, we can consider b as the molar volume of Xenon atoms, so we can equate it to the volume of one mole of Xe spheres. Let's write the expression for the b parameter as: \[b = \frac{4}{3} \pi r^3 N_A\] where \(N_A\) is Avogadro's number (number of molecules per mole).
02

Calculate the radius of a Xe atom using the given b parameter value

We know the van der Waals \(b\) parameter for Xe. Let's rearrange the equation derived in step 1 to solve for the radius (r) of a Xe atom: \[r^3 = \frac{3 b}{4 \pi N_A}\] We can now plug in the values for \(b\) and \(N_A\) (Avogadro's number: \(6.022 \times 10^{23}\, \text{mol}^{-1}\)) and calculate the radius of a Xe atom: \[r = \sqrt[3]{\frac{3 b}{4 \pi N_A}}\]
03

Compare the calculated radius with the given value

After calculating the radius of a Xe atom using the formula from Step 2, we can compare it with the given value of \(1.30 \unicode{x212B}\) A. If the calculated radius is close to the given value, it confirms that the van der Waals b parameter can be used to determine the size of atoms or molecules.

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