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Chapter 2: Problem 13

Calculate the force of attraction between a \(\mathrm{Ca}^{2+}\) and an \(\mathrm{O}^{2-}\) ion the centers of which are separated by a distance of \(1.25 \mathrm{nm}\)

Short Answer

Expert verified
Answer: The electrostatic force of attraction between the calcium and oxygen ions is approximately -1.22 x 10⁻⁹ N.

Step by step solution

01

Identify charges and distance

The calcium ion has a charge of +2 (Ca²⁺) and the oxygen ion has a charge of -2 (O²⁻). The distance separating them is 1.25 nm. Step 2: Convert the distance from nanometers to meters
02

Convert distance to meters

The distance is given in nanometers (nm), but we need to convert it to meters for our calculation. One nanometer is equal to 1.0 x 10⁻⁹ meters, so multiply the given distance by this conversion factor: \(1.25 \mathrm{nm} \times (1.0 \times 10^{-9}) \mathrm{m/nm} = 1.25 \times 10^{-9} \mathrm{m}\). Step 3: Calculate charges in Coulombs
03

Calculate charges in Coulombs

We need to convert the charges from their ion notation to Coulombs. The elementary charge is approximately 1.602 x 10⁻¹⁹ Coulombs, so multiply the charge of each ion by this value: \((+2 \times 1.602 \times 10^{-19} \mathrm{C})\) for calcium and \((-2 \times 1.602 \times 10^{-19} \mathrm{C})\) for oxygen. Step 4: Apply Coulomb's Law
04

Apply Coulomb's Law

Now we can use Coulomb's Law to calculate the force of attraction between the ions. Coulomb's Law is expressed as: \(F = k\frac{q_1q_2}{r^2}\), where F is the force between the charges, k is the electrostatic constant (\(8.9875 \times 10^9 \mathrm{Nm^2/C^2}\)), \(q_1\) and \(q_2\) are the charges, and r is the distance between the charges. Plug in the values for the charges and the distance: \(F = (8.9875 \times 10^9) \frac{(+2 \times 1.602 \times 10^{-19})(-2 \times 1.602 \times 10^{-19})}{(1.25 \times 10^{-9})^2}\). Step 5: Calculate the force
05

Calculate the force

Perform the calculation from Step 4 and then round the result to an appropriate number of significant figures: \(F \approx -1.22 \times 10^{-9} \mathrm{N}\). The negative sign indicates that the force between the ions is attractive, which is expected for oppositely charged particles.

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Most popular questions from this chapter

Relative to electrons and electron states what does each of the four quantum numbers specify?

(a) What electron subshell is being filled for the rare earth series of elements on the periodic table? (b) What electron subshell is being filled for the actinide series?

The net potential energy between two adjacent ions, \(E_{N},\) may be represented by the sum of Equations 2.8 and \(2.9 ;\) that is, $$E_{N}=-\frac{A}{r}+\frac{B}{r^{n}}$$ Calculate the bonding energy \(E_{0}\) in terms of the parameters \(A, B,\) and \(n\) using the following procedure: 1\. Differentiate \(E_{N}\) with respect to \(r,\) and then set the resulting expression equal to zero, since the curve of \(E_{N}\) versus \(r\) is a minimum at \(E_{0}\) 2\. Solve for \(r\) in terms of \(A, B,\) and \(n,\) which yields \(r_{0},\) the equilibrium interionic spacing. 3\. Determine the expression for \(E_{0}\) by substitution of \(r_{0}\) into Equation 2.11

Allowed values for the quantum numbers of electrons are as follows: \\[ \begin{aligned} n &=1,2,3, \ldots \\ l &=0,1,2,3, \ldots, n-1 \\ m_{l} &=0,\pm 1,\pm 2,\pm 3, \ldots, \pm l \\ m_{s} &=\pm \frac{1}{2} \end{aligned} \\] The relationships between \(n\) and the shell designations are noted in Table \(2.1 .\) Relative to the subshells, \(l=0\) corresponds to an \(s\) subshell \(l=1\) corresponds to a \(p\) subshell \(l=2\) corresponds to a \(d\) subshell \(l=3\) corresponds to an \(f\) subshell For the \(K\) shell, the four quantum numbers for each of the two electrons in the 1 s state in the order of \(n l m_{l} m_{s},\) are \(100\left(\frac{1}{2}\right)\) and \(100\left(-\frac{1}{2}\right) .\) Write the four quantum numbers for all of the electrons in the \(L\) and \(M\) shells, and note which correspond to the \(s, p,\) and \(d\) subshells.

Without consulting Figure 2.6 or Table \(2.2,\) determine whether each of the electron configurations given below is an inert gas, a halogen, an alkali metal, an alkaline earth metal, or a transition metal. Justify your choices. (a) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{5}\) (b) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{7} 4 s^{2}\) (c) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6}\) (d) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{1}\) (e) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 3 d^{10} 4 s^{2} 4 p^{6} 4 d^{5} 5 s^{2}\) (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2}\)
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