For \(\mathrm{NaCl}\), the ionic radii for \(\mathrm{Na}^{+}\)and \(\mathrm{Cl}^{-}\) ions are \(0.102\) and \(0.181 \mathrm{~nm}\), respectively. If an externally applied electric field produces a \(5 \%\) expansion of the lattice, compute the dipole moment for each \(\mathrm{Na}^{+}-\mathrm{Cl}^{-}\)pair. Assume that this material is completely unpolarized in the absence of an electric field.

Short Answer

Expert verified
Based on the given information, calculate the dipole moment for each Na+ - Cl- pair after a 5% expansion of the lattice due to an external electric field. Answer: The dipole moment for each Na+ - Cl- pair after a 5% expansion of the lattice due to an external electric field is approximately 4.76 × 10⁻³⁰ C·m.

Step by step solution

01

Calculate the initial distance between the ions Na+ and Cl-

The ionic radii for Na+ and Cl- ions are given as 0.102 and 0.181 nm, respectively. The initial distance between the ions can be found by summing up their radii: Initial distance (d) = Radius of Na+ + Radius of Cl- = 0.102 nm + 0.181 nm = 0.283 nm
02

Calculate the new distance between the ions after a 5% expansion

Due to the external electric field, the lattice expands by 5%. To find the new distance between the ions, we need to add 5% to the initial distance: New distance (d') = Initial distance × (1 + 5%) = 0.283 nm × (1 + 0.05) = 0.283 nm × 1.05 = 0.29715 nm
03

Calculate the charge of the ions

For an Na+ ion, the charge is +1 elementary charge. For a Cl- ion, the charge is -1 elementary charge. The elementary charge is approximately 1.6 × 10⁻¹⁹ C. Since the charges have the same magnitude and opposite signs, we can use the absolute value of the charge for the calculations: Charge (q) = 1 × 1.6 × 10⁻¹⁹ C = 1.6 × 10⁻¹⁹ C
04

Calculate the dipole moment

The dipole moment can be calculated using the formula: Dipole moment (p) = charge × distance We will use the new distance (d') between the ions after the 5% expansion and the charge of the ions to calculate the dipole moment: Dipole moment (p) = 1.6 × 10⁻¹⁹ C × 0.29715 nm Note that we need to convert the distance from nanometers to meters. The conversion factor is 1 nm = 10⁻⁹ m: Dipole moment (p) = 1.6 × 10⁻¹⁹ C × (0.29715 × 10⁻⁹ m) = 4.7564 × 10⁻³⁰ C·m Thus, the dipole moment for each Na+ - Cl- pair, under an externally applied electric field that produces a 5% expansion of the lattice, is approximately 4.76 × 10⁻³⁰ C·m.

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