Concept explainers
Normalize the following wavefunctions over the range indicated. You may have to use the integral table in Appendix 1.
(a)
(c)
(e)
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Chapter 10 Solutions
Student Solutions Manual for Ball's Physical Chemistry, 2nd
- 8C.4 (a) the moment of inertia of a CH4 molecule is 5.27 x 10^-47 kg m^2. What is the minimum energy needed to start it rotating? 8C.5 (a) use the data in 8C.4 (a) to calculate the energy needed excite a CH4 molecule from a state with l=1 to a state with l=2arrow_forwardfrom x =0 tox = L. (b) What are the Si units of this unnormalized fur 2. (a) Determine the normalization constant for the particle in a box atypical wave function b which equais NyaL - 2 in the box from x-C tox=L and equals zero outside the box. You'll need to solve the integral below. (b) Explain how this function does (or does not) satisty the boum conditions for a particle in a box. 1= *dz here if the narticle is an electron, the sphere has a radlusarrow_forwardBohr’s model can be used for hydrogen-like ions—ions thathave only one electron, such as He + and Li2+ . (a) Why isthe Bohr model applicable to He + ions but not to neutral Heatoms? (b) The ground-state energies of H, He + , and Li2 + aretabulated as follows: By examining these numbers, propose a relationship betweenthe ground-state energy of hydrogen-like systems and thenuclear charge, Z. (c) Use the relationship you derive in part(b) to predict the ground-state energy of the C5+ ion.arrow_forward
- Bohr’s model can be used for hydrogen-like ions—ions thathave only one electron, such as He + and Li2 + . (a) Why isthe Bohr model applicable to He + ions but not to neutral Heatoms? (b) The ground-state energies of H, He + , and Li2 + aretabulated as follows:By examining these numbers, propose a relationship betweenthe ground-state energy of hydrogen-like systems and thenuclear charge, Z. (c) Use the relationship you derive in part(b) to predict the ground-state energy of the C5 + ion.arrow_forwardImagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In each case, give your reasons for accepting or rejecting each function. (1) Þ(x) = x²; (iv) y(x) = x 5. (ii) ¥(x) = ; (v) (x) = e-* ; (iii) µ(x) = e-x²; (vi) p(x) = sinxarrow_forwardSuppose that 1.0 mol of perfect gas molecules all occupy the lowest energy level of a cubic box. (a) How much work must be done to change the volume of the box by ΔV? (b) Would the work be different if the molecules all occupied a state n ≠ 1? (c) What is the relevance of this discussion to the expression for the expansion work discussed in Topic 2A? (d) Can you identify a distinction between adiabatic and isothermal expansion?arrow_forward
- Imagine a particle free to move in the x direction. Which of the following wavefunctions would be acceptable for such a particle? In eachcase, give your reasons for accepting or rejecting each function. (i) Ψ(x)=x2; (ii) Ψ(x)=1/x; (iii) Ψ(x)=e-x^2.arrow_forwardA normalized wavefunction for a particle confined between 0 and L in the x direction is ψ = (2/L)1/2 sin(πx/L). Suppose that L = 10.0 nm. Calculate the probability that the particle is (a) between x = 4.95 nm and 5.05 nm, (b) between x = 1.95 nm and 2.05 nm, (c) between x = 9.90 nm and 10.00 nm, (d) between x = 5.00 nm and 10.00 nm.arrow_forwardConsider a particle of mass m confined to a one-dimensional box of length L and in a state with normalized wavefunction ψn. (a) Without evaluating any integrals, explain why ⟨x⟩ = L/2. (b) Without evaluating any integrals, explain why ⟨px⟩ = 0. (c) Derive an expression for ⟨x2⟩ (the necessary integrals will be found in the Resource section). (d) For a particle in a box the energy is given by En = n2h2/8mL2 and, because the potential energy is zero, all of this energy is kinetic. Use this observation and, without evaluating any integrals, explain why <p2x> = n2h2/4L2.arrow_forward
- (a) If  = 3x? and B = , then show that  and ß donot commute with respect to the function f(x) = sin x. Show, if the wave function, w) = A cos(kx) + iA sin(kx) is an Eigen-function of the linear momentum operator, P and if so, what is the Eigen value. (Note: A and k are constants). (b)arrow_forward(1) (2) (b) Look very carefully at the picture below. Give the relevant quantum numbers. Explain your answer. y-axis (c) (1) What is a wavefunction? (ii) What are the two parts of a wavefunction?arrow_forward8C.5 (a) use the data in 8C.4 (a) to calculate the energy needed excite a CH4 molecule from a state with l=1 to a state with l=2arrow_forward
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