Physical Chemistry Answers

A particle is in a one-dimensional potential of: V(x)= λx4 , with λ>0. The upper bound of the ground state can be calculated using the variational method. The following trial function may be used: 1 2 2 ( ) exp{ } 2 a φ x N ax = − where Na is the normalization factor. (1) [5 pts] Does this trial function satisfy the requirements of the symmetry and the asymptotic behavior/boundary conditions of the ground-state wavefunction? Explain why. (2) [5 pts] Find Na. (You can use integral equations in McQuarrie or any other textbooks directly.) (3) [5 pts] Compute the upper bound of the ground state using this trial function. You don’t need to calculate the integrals analytically or numerically. Keep the integrals in your answer.

In Chem 465, we learn that the Hamiltonian of a rigid rotor is H=BJ2 where B is the rotational constant and J is the rotational angular momentum. The rotational energies are: E=BJ(J+1) 5 where J=0,1,2… is the rotational angular momentum quantum number. Consider a perturbation: 𝐻𝐻’ = 1 2 𝜆𝜆�𝐽𝐽̂+ + 𝐽𝐽̂−�. where λ is a constant. The operators 𝐽𝐽̂+ and 𝐽𝐽̂− have the following effects on the spherical harmonics 𝑌𝑌𝐽𝐽 𝑚𝑚𝐽𝐽 : 𝐽𝐽̂−𝑌𝑌𝐽𝐽 𝑚𝑚𝐽𝐽 = 𝐽𝐽𝑌𝑌𝐽𝐽−1 𝑚𝑚𝐽𝐽−1 , 𝐽𝐽̂−𝑌𝑌0 0 = 0 𝐽𝐽̂+𝑌𝑌𝐽𝐽 𝑚𝑚𝐽𝐽 = (𝐽𝐽 + 1)𝑌𝑌𝐽𝐽+1 𝑚𝑚𝐽𝐽+1 (1) [5 pts] Prove that the first-order perturbation vanishes based on symmetry arguments. (2) [10 pts] For the ground rotational state (J=0), compute the second-order contribution to its energy due to the perturbation by the J=1 state. (3) [3 pts] For the ground rotational state (J=0), prove that the second-order contribution to its energy due to the perturbation by the J=2 state vanishes

] The 1 H nucleus (i.e., the proton) has a nuclear spin of I=1/2. In the H2 molecule, the coupling of the two nuclear spins is simular to that of the two electrons in the helium atom, i.e., the couplng can be symmetric or anti-symetric with respect to the exchange of the two nuclei. The 1 H nucleus, as well as the electron, is a fermion, i.e., the total wavefuction of the H2 molecule changes its sign if the two 1 H muclei are exchanged. In its ground vibronic (vibrational-electronic) state [ 1Σ𝑔𝑔 + (v=0)], both the vibrational and the electronic wavefunctions are symmetric with respect to the exchange of the two nuclei. Use the information provided above and what you learned in Chem 465, explain why the natural abundance of ortho-H2 and para-H2 is 3:1. The ortho-H2 molecules are those on the J=1,3,5… rotational levels, while para-H2 molecules are those on the J=0,2,4… rotational levels

1. In Bohr’s model for the hydrogen atom, the field-free energy of an orbital is determined by: (a) the principal quantum number n. (b) the orbital angular momentum quantum number l. (c) both n and l. (d) l and its projection, i.e., the magnetic quantum number ml. 2. In Bohr’s model, the kinetic energy Ek and the potential energy Ep of the hydrogen atom have the following relation: (a) Ek = 2Ep. (b) Ek = ½Ep. (c) Ek = -2Ep. (d) Ek =-½Ep. 3. The angular parts of the wavefunctions of a 3D spherical potential well with an infinite depth, the hydrogen atom, and a rigid rotor are the same, which is: (e) a sine function. (f) a Hermite polynomial. (g) a Legendre polynomial. (h) a spherical harmonics

The variational method provides an upper bound to the ground-state energy of a system. However, it may also be used to calculate first excited state energies if: (a) the trial function has the symmetry of the first excited state, which is different from the ground state. (b) the trial function has the boundary conditions of the first excited state, which are different from the ground state. (c) the trial function has the asymptotic behavers of the first excited state, which are different from the ground state. (d) the trial function has the minimum and maximum values of the first excited state, which are different from the ground state

Calculate the hardness of a water Sample whose 20 ml required 30 ml EDTA, 10ml of standard

calcium chloride solution, whose strength is equivalent to 300mg of CaCO3 per 200 ml required 20ml

of same EDTA solution.

 In order for a chemical equation to be properly balanced, what must be true?

seperation of charge into a more positive area and a more negative area creates

How to plot graph according to integrated rate equation of zero or 1st order of reaction from a given set of time and concerntration values

The probability of finding the harmonic oscillator in the energy level n=1 is?(niglate Zpe ) assume hw=kBT