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physical chemistry
Physical Chemistry 3rd edition Thomas Engel, Philip Reid - Solutions
Solids generally expand as the temperature increases. Such an expansion results from an increase in the bond length between adjacent atoms as the vibrational amplitude increases. Will a harmonic potential lead to thermal expansion? Will a Morse potential lead to thermal expansion?
Why would you observe a pure rotational spectrum in the microwave region and a rotational– vibrational spectrum rather than a pure vibrational spectrum in the infrared region?
Calculate the position of the center of mass of a. 1H19F, which has a bond length of 91.68 pm;b. HD, which has a bond length of 74.15 pm.
An 1H35Cl molecule has the rotational quantum number J = 8 and vibrational quantum number n = 0.a. Calculate the rotational and vibrational energy of the molecule. Compare each of these energies with kBT at 300. K.b. Calculate the period for vibration and rotation. How many times does the molecule
Calculate the first five energy levels for a 35Cl2 molecule which has a bond length of 198.8 pm if ita. Rotates freely in three dimensionsb. Is adsorbed on a surface and forced to rotate in two dimensions.
Calculate the constants b1 and b2 in Equation (18.9) for the condition x (0) = xmax, the maximum extension of the oscillator. What is v (0) for this condition?
Calculate the reduced mass, the moment of inertia, the angular momentum, and the energy in the J = 1 rotational level for H2 which has a bond length of 74.14 pm.
Using your results for Problems P 18.11, 17, 29, and 32, calculate the uncertainties in the position and momentum σ2p = 〈p2〉 − 〈p〉2 σ2x = 〈x2〉 − 〈x〉2 for the ground state (n = 0) and first two excited states (n = 1 and n = 2) of the quantum
Evaluate 〈x〉, for the ground state (n = 0) and first two excited states (n = 1 and n = 2) of the quantum harmonic oscillator.
Use √(x2) as calculated in Problem P18.17 as a measure of the vibrational amplitude for a molecule. What fraction is of the 127.5-pm bond length of the 1H35Cl molecule for n = 0, 1, and 2? The force constant for H35Cl is 516 N m−1.
By substituting in the Schrödinger equation for rotation in three dimensions, show that the rotational wave function (5/16π)1/2 (3cos2θ − 1) is an eigenfunction of the total energy operator. Determine the energy eigenvalue.
Evaluate the average linear momentum of the quantum harmonic oscillator, 〈 px 〉, for the ground state (n = 0) and first two excited states (n = 1and n = 2).
By substituting in the Schrödinger equation for the harmonic oscillator, shows that the ground-state vibrational wave function is an eigenfunction of the total energy operator. Determine the energy eigenvalue.
Evaluate the average kinetic and potential energies, 〈Ekinetic〉 and 〈Epotential〉, for the second excited state (n = 2) of the harmonic oscillator by carrying out the appropriate integrations.
Verify that ψ1(x) in Equation 18.31 is a solution of the Schrödinger equation for the quantum harmonic oscillator. Determine the energy eigenvalue.
A 1H19F molecule, with a bond length of 91.68 pm, absorbed on a surface rotates in two dimensions.a. Calculate the zero point energy associated with this rotation.b. What is the smallest quantum of energy that can be absorbed by this molecule in a rotational excitation?
At 300 K, most molecules are not in their ground rotational state. Is this also true for their vibrational degree of freedom? Calculate Nn=1 Nn=0 and Nn=2 Nn=0 for the 127I2 molecule. Make sure that you take the degeneracy of the levels into account for the 127I2 molecule for which the force
The force constant for a 1H127I molecule is 314 N m−1.a. Calculate the zero point vibrational energy for this molecule for a harmonic potential.b. Calculate the light frequency needed to excite this molecule from the ground state to the first excited state.
The force constant for the35Cl2molecule is 323 N mˆ’1. Calculate the vibrational zero point energy of this molecule. If this amount of energy were converted to translational energy, how fast would the molecule be moving? Compare this speed to the root mean square speed from the kinetic
Is it possible to simultaneously know the angular orientation of a molecule rotating in a two-dimensional space and its angular momentum? Answer this question by evaluating the commutator [∅, − ih(∂/∂∅]
Show that the function Y20 (θ, ∅) = (5/16π)1/2 (3cos2 θ − 1) is normalized over the interval 0 ≤ θ ≤ π and 0 ≤ ∅ ≤ 2π.
A coin with a mass of 8.31g suspended on a rubber band has a vibrational frequency of 7.50 s−1.Calculatea. The force constant of the rubber bandb. The zero point energy,c. The total vibrational energy if the maximum displacement is 0.725 cmd. The vibrational quantum number corresponding to the
Evaluate 〈x2〉 for the ground state (n = 0) and first two excited states (n = 1 and n = 2) of the quantum harmonic oscillator.
The vibrational frequency of 35Cl2 is 1.68 × 1013 s−1. Calculate the force constant of the molecule. How large a mass would be required to stretch a classical spring with this force constant by 2.25 cm? Use the gravitational acceleration on Earth at sea level for this problem.
Evaluate the average kinetic and potential energies, 〈Ekinetic〉 and 〈Epotential〉, for the ground state n = 0 of the harmonic oscillator by carrying out the appropriate integrations.
Calculate the frequency and wavelength of the radiation absorbed when a quantum harmonic oscillator with a frequency of 3.15 × 1013 s−1 makes a transition from the n = 2 to the n = 3 state.
Two 3.25-g masses are attached by a spring with a force constant of k = 450. kg s−2. Calculate the zero point energy of the system and compare it with the thermal energy kBT at 298 K. If the zero point energy were converted to translational energy, what would be the speed of the masses?
Show by carrying out the appropriate integration that the total energy eigenfunctions for the harmonic oscillator ψ0 (x) (α /π)1/4 e-(1/2)ax2 and ψ2(x) (α /4π)1/4 (2αx - 1) e-(1/2)ax2 are orthogonal over the interval −∞ < x < ∞ and that ψ2(x) is normalized over the same
Evaluate the average of the square of the linear momentum of the quantum harmonic oscillator, 〈px2〉 for the ground state (n = 0) and first two excited states (n = 1and n = 2).
Show by carrying out the necessary integration that the eigenfunctions of the Schrodinger equation for rotation in two dimensions, 1/√2π eiml∅ and 1/√2π einl∅, ml ≠ nl are orthogonal.
In discussing molecular rotation, the quantum number J is used rather than l. Calculate Erot / kBT for 1H81Br for J = 0, 5, 10, and 20 at 298 K. The bond length is 141.4 pm. For which of these values of J is Erot / kBT ≥ 10?
The vibrational frequency for D2 expressed in wave numbers is 3115 cm−1. What is the force constant associated with the D⎯D bond? How much would a classical spring with this force constant be elongated if a mass of 1.50 kg were attached to it? Use the gravitational acceleration on Earth at sea
At what values of θ does Y20 (θ, ∅) = (5/16π)1/2 (3cos2θ − 1) have nodes? Are the nodes points, lines, planes, or other surfaces?
The wave functions px and dxz are linear combinations of the spherical harmonic functions, which are eigenfunctions of the operators Ĥ, l̂2, and l̂z for rotation in three dimensions. The combinations have been chosen to yield real functions. Are these functions still eigenfunctions of l̂z?
1H19F has a force constant of 966 N m−1 and a bond length of 91.68 pm. Calculate the frequency of the light corresponding to the lowest energy pure vibrational and pure rotational transitions. In what regions of the electromagnetic spectrum do the transitions lie?
Draw a picture (to scale) showing all angular momentum cones consistent with l = 5. Calculate the half angles for each of the cones.
In discussing molecular rotation, the quantum number J is used rather than l. Using the Boltzmann distribution, calculate nJ /n0 for 1H35Cl for J = 0, 5, 10, and 20 at T = 1025 K. Does nJ /n0 go through a maximum as J increases? If so, what can you say about the value of J corresponding to the
In this problem you will derive the commutator [l̂x, l̂y] = ihl̂z.a. The angular momentum vector in three dimensions has the form l = ilx + jly + klz, where the unit vectors in the x, y, and z directions are denoted by i, j, and k. Determine lx , ly , and lz by expanding the 3 × 3 cross product
A gas-phase 1H127I molecule, with a bond length of 160.92 pm, rotates in three dimensional space.a. Calculate the zero point energy associated with this rotation.b. What is the smallest quantum of energy that can be absorbed by this molecule in a rotational excitation?
For a two-dimensional harmonic oscillator, V (x, y) = kxx2 + kyy2 Write an expression for the energy levels of such an oscillator in terms of kx and ky.
What is the degeneracy of the energy levels for the rigid rotor in two dimensions? If it is not 1, explain why.
Conservation of energy requires that the variation of the potential and kinetic energies with the oscillator extension be exactly out of phase. Explain this statement.
Use the anharmonic potential function in Figure 18.7 to demonstrate that rotation and vibration are not separable degrees of freedom for large quantum numbers.Figure 18.7 (X)A
Figure 18.12 shows the solutions to the time-independent Schrödinger equation for the rigid rotor in two dimensions. Describe the corresponding solutions for the time-dependent Schrödinger equation.Figure 18.12 m = 1 m,=2 m = 3 m = 4 m, = 5 m,= 6
The zero point energy of the particle in the box goes to zero as the length of the box approaches infinity. What is the appropriate analogue for the quantum harmonic oscillator?
How are the spherical harmonics combined to form real p and d functions? What is the advantage in doing so?
What makes the z direction special such that l̂2, Ĥ, and l̂z commute, whereas l̂2, Ĥ, and l̂x do not commute?
Why is only one quantum number needed to characterize the eigenfunctions for rotation in two dimensions, whereas two quantum numbers are needed to characterize the eigenfunctions for rotation in three dimensions?
The two linearly independent total energy eigenfunctions for rotation in two dimensions areWhat is the evolution in time of ˆ… for each of these solutions? 1 Ф, (0) 1 ·e and o_(0) 2T |-4 1,
What is the difference between a bit and a qubit?
Why is it possible to write the total energy eigenfunctions for rotation in three dimensions in the form Y (θ, ∅) = Θ (θ) Φ (∅)?
Explain why the amplitude of the total energy eigenfunctions for the quantum mechanical harmonic oscillator increase with ˆ£xˆ£ as shown in Figure 18.10.Figure 18.10
How does the total energy of the quantum harmonic oscillator depend on its maximum extension?
Why can the angular momentum vector lie on the z axis for two-dimensional rotation in the xy plane but not for rotation in three-dimensional space?
Does the bond length of a real molecule depend on its energy? Answer this question by referring to Figure 18.7. The bond length is the midpoint of the horizontal line connecting the two parts of V (x).Figure 18.7 FX
Does the average length of a quantum harmonic oscillator depend on its energy? Answer this question by referring to the harmonic potential function shown in Figure 18.7. The average length is the midpoint of the horizontal line connecting the two parts of V (x).Figure 18.7 FX
Spatial quantization was discussed in Section 18.8. Suppose that we have a gas consisting of atoms, each of which has a nonzero angular momentum. Are all of their angular momentum vectors aligned?
Are the real functions listed in Equations (18.62) and (18.63) eigenfunctions of lz? Justify your answer.
Why does the energy of a rotating molecule depend on l, but not on ml?
Why is the probability of finding the harmonic oscillator at its maximum extension or compression larger than that for finding it at its rest position?
The muzzle velocity of a rifle bullet is 890.m s−1 along the direction of motion. If the bullet weighs 35 g, and the uncertainty in its momentum is 0.20%, how accurately can the position of the bullet be measured along the direction of motion?
For linear operators Aˆ, Bˆ, and Cˆ, show that [Aˆ,BˆCˆ] = [Aˆ,Bˆ]Cˆ + Bˆ[Aˆ,Cˆ].
What is wrong with the following argument? We know that the functions ψ n (x) = √2/a sin(nπ x/a) are eigenfunctions of the total energy operator for the particle in the infinitely deep box. We also know that in the box, E = P2x/2m + V(x) = P2x/2m. Therefore, the operator for E total is
Evaluate the commutator [ xˆ, Pˆ 2x] by applying the operators to an arbitrary function f (x).
Evaluate the commutator [(d2 /dy2) − y,(d2/dy2) + y] by applying the operators to an arbitrary function f (y).
In this problem, you will carry out the calculations that describe the Stern€“Gerlach experiment shown in Figure 17.2. Classically, a magnetic dipole μ has the potential energy E = ˆ’μ €¢ B. If the field has a gradient in the z direction,
Evaluate the commutator [xˆ, pˆ x] by applying the operators to an arbitrary function f (x). What value does the commutator [ pˆ x , xˆ] have?
Evaluate the commentator [d/dx, x2] by applying the operators to an arbitrary function f (x).
Apply the Heisenberg uncertainty principle to estimate the zero point energy for the particle in the box. a. First justify the assumption that Δ x ≤ a and that, as a result, Δp ≥ h/2a. Justify the statement that, if Δp ≥ 0, we cannot know that E = p2 /2m is identically zero.b.
If the wave function describing a system is not an eigenfunction of the operator Bˆ, measurements on identically prepared systems will give different results. The variance of this set of results is defined in error analysis as σ2B = 〈(B − 〈B〉2〉 where B is the value of the
Evaluate the commentator [(d2 /dy2), y] by applying the operators to an arbitrary function f (x).
Revisit the TV picture tube of Example Problem 17.3. Keeping all other parameters the same, what electron energy would result in a position uncertainty of 1.00 × 10−8 m?
Evaluate the commentator [y2, d /dy2] by applying the operators to an arbitrary function f ( y).
Revisit the double-slit experiment of Example Problem 17.2. Using the same geometry and relative uncertainty in the momentum, what electron momentum would give a position uncertainty of 2.50 × 10−10 m? What is the ratio of the wavelength and the slit spacing for this momentum? Would you expect a
Evaluate the commentator [Pˆx + Pˆ 2x, Pˆ2x] by applying the operators to an arbitrary function f (x).
Consider the entangled wave function for two photons,Assume that the polarization operator Pˆi has the properties Pˆψi i (H) = ˆ’ψ i (H) and Pˆiψ i (V ) = + ψ i (V ), where i = 1 or i = 2. H and V designate
Evaluate [Â, B̂] if  = x + d/x and B̂ = x – d/dx ?
Evaluate the commentator [x(∂/∂y), y(∂/∂x)] by applying the operators to an arbitrary function f (x, y).
Another important uncertainty principle is encountered in time-dependent systems. It relates the lifetime of a state Δt with the measured spread in the photon energy ΔE associated with the decay of this state to a stationary state of the system. “Derive” the relation ΔE Δt ≥ h/2 in
a. Show that ψ(x) = e−x2/2 is an eigenfunction of Aˆ = x2 − ∂2/∂x2b. Show that B̂ ψ(x), (where Bˆ = x − ∂/∂x) is another eigenfunction of Aˆ.
Evaluate the commentator [d /dy,1/y2] by applying the operators to an arbitrary function f (r).
Consider the results of Figure 17.5 more quantitatively. Describe the values of x and k by x ± Δx and k0± Δk. Evaluate Δx from the zero of distance to the point at which the envelope of ψ*(x)ψ (x) is reduced to one-half of its
In this problem, we consider the calculations for σpand σxfor the particle in the box shown in Figure 17.5 in more detail. In particular, we want to determine how the absolute uncertainty Δpxand the relative uncertainty Δpx/pxof a single peak
Describe the trends in Figure 17.5 that you expect to see as the quantum number n increases.Figure 17.5 n-101 -6 -4 -2 k/(1010 n-15 -1.0 -0.5 0.5 1.0 k(1010 m ) n-5 -0.6 -0.4 -0.2 0.2 0.4 0.6 k/(1010 m ') A. n-1 -0.3 -0.2 -0.1 0.2 0.3 0.1 k(1010 m 1) ---------
An electron and a He atom have the same uncertainty in their speed. What can you say about the relative uncertainty in position for the two particles?
Discuss whether the results shown in Figure 17.7 are consistent with local realism.Figure 17.7 300 250 200 150 100 50 -8 -6 -4 -2 4 6. 8. Detector 2 position/mm Coincidence counts
How would the results of Stern and Gerlach be different if they had used a homogeneous magnetic field instead of an inhomogeneous field?
How would the results of Stern and Gerlach be different if they had used a Mg beam instead of an Ag beam?
Why is √(p2) rather than 〈 p〉 used to calculate the relative uncertainty for the particle in the box?
Explain the following statement: If h = 0, it would be possible to measure the position and momentum of a particle exactly and simultaneously.
Why isn’t the motion of a human being described by the Schrödinger equation rather than Newton’s second law if every atom in our body is described by quantum mechanics?
The Heisenberg uncertainty principle says that the momentum and position of a particle cannot be known simultaneously and exactly. Can that information be obtained by measuring the momentum and quickly following up with a measurement of the position?
Which result of the Stern–Gerlach experiment leads to the conclusion that the operators for the z and x components of the magnetic moment do not commute?
Why does it follow from the Heisenberg uncertainty principle that it is not possible to make exact copies of quantum mechanical objects?
How does a study of the eigenfunctions for the particle in the box lead to the conclusion that the position uncertainty has its minimum value for n = 1?
Derive a relationship between [Â, B̂] and [B̂, Â]
Why is the statistical concept of variance a good measure of uncertainty in a quantum mechanical measurement?
Why does the relative uncertainty in x for the particle in the box increase as n → ∞?
Why is it not possible to reconstruct the wave function of a quantum mechanical superposition state from experiments?
Why is maintaining the entanglement of pairs A and B and A and X the crucial ingredient of teleportation?
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