3. The potential energy of two particles in the electrostatic field when they are separated by a distance r is given by α В U = - + r r8 where a and ẞ are constants. At what separation will they form a stable compound?
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- 1. A model for the potential energy interaction between the two nitrogen atoms in the N2 molecule is proposed that has the form: v(4) = 4E, (4)* - ()'] 12 a. Find the position of the potential minimum and its value there, in terms of o and Eo, respectively. What do these parameters represent physically about the molecule? b. Draw a hand sketch of V(r) showing rmin, V(rmin) and where V crosses the r axis. c. If the atom is disturbed from its equilibrium by a small amount, show that the 7.56. frequency of oscillation is w = d. For molecular nitrogen, the bond length is 1.1 x 10-1°m, the bond (binding) energy is 9.79 eV (15.66 x 10-19 J), the mass is 14 amu = 23.38 x 10-27 kg. In the spectroscopy laboratory, this vibration is measured to be 8.8 x 1013 Hz. Is this a good model for the interatomic potential? (Recall o = 2n times frequency in Hz) where m is the mass of a nitrogen atom.A + Given that B 7-10- == 13) Suppose that the interaction energy between two atoms is given by E (r) the atoms from a stable molecule with an internuclear distance of 0.3 nm and a dissociation energy of 4 eV, calculate A and B. Also calculate the force require to break the molecule and the critical distance between the nuclei for which this occurs.17) Shown below is a structure of seven atoms with a "B" atom in the middle surrounded by "A" atoms. Nearest neighbors are separated by ro. A A A B A A A: 42 x 10-2¹ J B: 7 x 10-21 J C: 30 x 10-21 J D: 15 x 10-21 J E: 5 x 10-21 J A How much energy is required to remove only the B atom from the center, given that the well depth for an A-A pair is 2 x 10-2¹J and for a A-B pair is 5 x 10-2¹J?
- 3. Show that in an ideal hexagonal-close-packed (hcp) structure, where the atomic spheres touch each other, the ratio c/a is given by :-()* ( 8 \ 1/2 = 1.633. (The hep structure is discussed in Section 7.) 4 The packing ratio is defined as the fraction of the total volume of the cell that is filled by atoms. Determine the maximum values of this ratio for equal spheres located at the points of simple-cubic, body-centered-cubic, and face-centered-cubic crystals.How many protons does a gold atom have: O 1 O 26 79 O 197 ype here to search 近4. Liner ionic crystal. Consider a line of 2N ions of alternating charge ±q with a repulsive potential energy A/R" between nearest neighbors. (a) Show that at the equilibrium separation (CGS) (b) Let the crystal be compressed so that R. → R. (1-8). Show that the work done in a compressing a unit length of the crystal has the leading term C8²/2, where 2 (n-1)q² In 2 (CGS) C = U(R): = 2Nq² In 2 (₁-1). Ro n Ro To obtain the results in SI, replace q² by q²/4л. Note: We should not expect to obtain this result from the expression for U(R), but we must use the complete expression for U(R).
- 4. Calculate the potential energy of the system of Na* and CI ions when they are at a distance of 4Å.In an H₂O molecule, the distance of the oxygen from the attached hydrogen atoms is 0.0916 nm, and the angle between the two O-H bonds is 105.8°. If the hydrogen nuclei of a water molecule are detached one after the other, how much minimum work is needed to dismantle a water molecule? (Assume that the end products after the dismantling will be one 02-ion and two H¹+ ions.)2. Use the electrostatic model to calculate the energy (in kJ mol") required to break an 0...H hydrogen bond in a vacuum (er = 1) and in water (er 80.0). Let the O...H distance be 170 pm, and partial charges on the H and O atoms be +0.42e and -0.84e, respectively.
- The repulsive force between Na+ and Cl- ions can be written as F = Be-T/R where B = 5.45 x 10-6 N and R = 0.321 Å. At equilibrium, this force is equal to the magnitude of the attractive force given by F Determine the value of r at equilibrium. e² 4π€or²ny If the bonding energy, in terms of (r), between two ions is (2-1), x and y are constant and n is an integer number. a) Find the repulsive and attractive forces between the two ions. b) If the bonding energy between the two ions at the equilibrium state is given as Find the smallest distance between the two ions to achieve the state of equilibrium. ny. 2) 1-nA2 molecule is considered in a simple way as two touching A atoms as depicted in the Figure. Does this arrangement have a lower energy than two separated A atoms? The radius r. of the A atom is 0.1 nm. Using the Virial Theorem, consider the following: a. Calculate the total electrostatic potential energy (PE) of all the charges when they are arranged as shown in Figure. Is this configuration energetically favorable? b. Given that in the isolated A-atom the PE is 2 ×(-15 eV), calculate the change in PE in going from two isolated A-atoms to the A2 molecule. Using the Virial theorem, find the change in the total energy. Nucleus Nucleus e---