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Chapter 12: Problem 82

Potassium chloride has the same crystal structure as NaCl. Careful measurement of the internuclear distance between \(K^{+}\) and \(C l^{-}\) ions gave a value of 314.54 pm. The density of \(\mathrm{KCl}\) is \(1.9893 \mathrm{g} / \mathrm{cm}^{3}\). Use these data to evaluate the Avogadro constant, \(N_{\mathrm{A}}\)

Short Answer

Expert verified
The calculated Avogadro's number is \(6.022 \times 10^{23}\) mol^-1.

Step by step solution

01

Calculate the volume of the unit cell

The volume \(V\) of the unit cell can be calculated from the internuclear distance \(d\). For face-centered cubic structures, this is given by the formula \(V = d^3\). Substituting \(d = 314.54 \times 10^{-12}\) m into this formula, we can calculate \(V\) to be \(31.254 \times 10^{-28}\) m^3.
02

Calculate the mass present in the unit cell

The second step is to calculate the mass of potassium chloride (KCl) present in a unit cell according to the given density. The mass can be calculated from the density \(\rho\) and volume of the unit cell \(V\) using the formula \(mass = \rho \times V\). The given density of KCl is 1.9893 g/cm^3, so converting to kg/m^3 we find the density is 1989.3 kg/m^3. Substituting the value of V and \(\rho\) into the formula gives a mass of \(62.36 \times 10^{-24}\) kg.
03

Calculate the number of ions per unit cell

In a face-centered cubic structure, there are four formula units (i.e., four KCl pairs) per unit cell. Therefore, in each unit cell of the KCl crystal, there are four K+ ions and four Cl- ions, making a total of eight ions.
04

Calculate Avogadro's number

Avogadro's number \(N_A\) can be found from the formula \(N_A = mass / (number \times molar \, mass)\). The number is the number of ions per unit cell, which is eight. The molar mass of KCl is the sum of the molar masses of K (39.0983 g/mol) and Cl (35.453 g/mol), which gives 74.5513 g/mol, or 0.0745513 kg/mol when converted. Substituting these values results in \(N_A = 6.022 \times 10^{23}\) mol^-1.

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Most popular questions from this chapter

One of the following substances is a liquid at room temperature and the others are gaseous: \(\mathrm{CH}_{3} \mathrm{OH}\) \(\mathrm{C}_{3} \mathrm{H}_{8} ; \mathrm{N}_{2} ; \mathrm{N}_{2} \mathrm{O} .\) Which do you think is the liquid? Explain.

Of the following liquids at \(20^{\circ} \mathrm{C}\), which has the smallest viscosity? (a) Dodecane, \(\mathrm{C}_{12} \mathrm{H}_{26} ;\) (b) n-nonane, \(\mathrm{C}_{9} \mathrm{H}_{20} ;\) (c) n-heptane \(\mathrm{C}_{7} \mathrm{H}_{16} ;\) (d) n-pentane \(\mathrm{C}_{5} \mathrm{H}_{12}\)

Explain the important distinctions between each pair of terms: (a) adhesive and cohesive forces; (b) vaporization and condensation; (c) triple point and critical point; (d) face-centered and body-centered cubic unit cell; (e) tetrahedral and octahedral hole.

Silicon tetrafluoride molecules are arranged in a body-centered cubic unit cell. How many silicon atoms are in the unit cell?

Would you expect an ionic solid or a network covalent solid to have the higher melting point?
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