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Consider a real system. Assume that a real wavefunction is a combination of two orthogonal functions such that
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Physical Chemistry
- Suppose 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_forwardA particle freely moving in one dimension x with 0 ≤ x ≤ ∞ is in a state described by the normalized wavefunction ψ(x) = a1/2e–ax/2, where a is a constant. Evaluate the expectation value of the position operator.arrow_forwardImagine 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_forward
- Calculate the value of ml for a proton constrained to rotate in a circle of radius 100 pm around a fixed point given that the rotational energy is equal to the classical average energy at 25 degrees C. (Mass of a proton = 1.6726 x 10^-27 kg, classical average energy=1/2kBT, where kBT is Boltzman constant = 1.30 x 10^ -23 J K^-1, and T is the temperature.)arrow_forwardThe wave function for the ground state of the harmonic oscillator is Vo(x) = Ce-[mw/(2ħ)]x² where C is an arbitrary constant, ħ is Planck's constant divided by 2π, m is the mass of the particle, W = ✓k/m, and k is the "spring constant" for the harmonic oscillator. Part A Normalize this wave function. What is the (positive) value of C once this wave function is normalized? You will need the formula Se -∞ Express your answer in terms of w, m, ħ, and T. ► View Available Hint(s) C = 17 ΑΣΦ xa Xh عات a √x vx 18 X> IXI -ax² X.10n X = ? wwwwwwwwww √. aarrow_forwardVibrations in the diatomic molecule CO can be approximated as a harmonic oscillator, where the angular frequency w = 6.505 x 1013 Hz and the reduced mass is equal to u = 1.14 × 10-27 kg. Assume the molecule is in its fırst excited vibrational state. Its vibrational wavefunction can then be written as V1 (x) = (4) /2a xe where a = . If we were to measure the bond length of the molecule, what is the most likely displacement from the equilibrium bond distance in the first excited vibrational state? Give your answer in Angstroms [Note: The equilibrium displacement in the Quantum harmonic oscillator corresponds to r = 0, ie the coordinate x measures displacement from equilibrium]arrow_forward
- Use the data in Exercise E7F.6(b) to calculate the energy needed to excite an SF6 molecule from a state with l = 2 to a state with l = 3.arrow_forward8C.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_forwardFor a certain harmonic oscillator of effective mass 2.88 × 10−25 kg, the difference in adjacent energy levels is 3.17 zJ. Calculate the force constant of the oscillator.arrow_forward
- Calculate the energy of the quantum involved in the excitation of (i) an electronic oscillation of period 2.50 fs, (ii) a molecular vibration of period 2.21 fs, (iii) a balance wheel of period 1.0 ms. Express the results in joules and kilojoules per mole.arrow_forwardThe moment of inertia of an SF6 molecule is 3.07 × 10−45 kg m2. What is the minimum energy needed to start it rotating?arrow_forwardCharrow_forward
- Principles of Modern ChemistryChemistryISBN:9781305079113Author:David W. Oxtoby, H. Pat Gillis, Laurie J. ButlerPublisher:Cengage LearningPhysical ChemistryChemistryISBN:9781133958437Author:Ball, David W. (david Warren), BAER, TomasPublisher:Wadsworth Cengage Learning,