Problem Set 7 key - Recitation PDF

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Problem Set 07 – Quantum Mechanics Chem 105 1. (a) Louis deBroglie correctly postulated that matter has wavelike properties. What was the deBroglie wavelength of the baseball (0.145 kg) thrown at 108.1 mph (48.325 m/s) by Nolan Ryan in 1974, the fastest pitch recorded? 6.626 ×10−34 J •s h =9.46 ×10−35 m = λ= mv m (.145 kg)( 48.325 ) s (b) Assuming the uncertainty in the measured speed was 5.5 x10-28 m/s, what was uncertainty in the ball’s position as it passed the hitter? Did the fact that the baseball had wavelike properties add significantly to the challenge of hitting that baseball? h ∆ x∙m∆u≥ 4π 6.626 x 10−34 Js h =6.6 x 10−7 m = ∆ x≥ −28 4 π ∙ m∆ u 5.5 x 10 m 4 π ∙ 0.145 kg ∙ s No, the uncertainty in the ball’s position is about the size of 1 wavelength of red light, so the human eye does not detect any uncertainty

2. 3.67 Calculate the wavelengths of the following objects: a) A muon (a subatomic particle with a mass of 1.884 × 10–28 kg) traveling at 325 m/s −34 6.626 ×10 J • s h = =1.08 ×10−8 m λ= mv m (1.884 ×10−28 kg)(325 ) s –31 b) Electrons (me = 9.10938 × 10 kg) moving at 4.05 × 106 m/s in an electron microscope 6.626 ×10−34 J • s h = =1.80 ×10−10 m λ= mv −31 6 m (9.10938 ×10 kg)( 4.05 ×10 ) s c) An 82 kg sprinter running at 9.9 m/s 6.626× 10−34 J • s =8.2 ×10−37 m h = λ= mv m (82 kg)(9.9 ) s d) Earth (mass = 6.0 × 1024 kg) moving through space at 3.0 × 104 m/s 6.626 × 10−34 J • s h = =3.7 × 10−63 m λ= mv 24 4 m (6.0 ×10 kg)( 3.0 × 10 ) s

3. A synchrotron is a type of particle accelerator in which a ring of variable electric and magnetic fields accelerates particles (electrons, protons, etc.) to speeds nearing the speed of light. The largest synchrotron in the world is the Large Hadron Collider in Geneva, Switzerland where protons (mp = 1.6726x10-27 kg) can be accelerated to speeds of 0.999999991 c.

(a) What is the de Broglie wavelength of one of these protons? Protons: 6.626 ×10−34 J •s h −15 = =1.321 × 10 m λ= mv m (1.6726 ×10−27 kg)( 0.999999991 ×2.998 × 108 ) s (b) What is the wavelength of an electron (me = 9.109x10–31 kg) moving at that same speed? (not considering relativistic effects) Electrons: 6.626× 10−34 J • s h =2.426 ×10−12 m = λ= mv −31 8m (9.10938 ×10 kg)( 0.999999991 ×2.998 × 10 ) s (c) If the relative uncertainty in the velocity of a proton in the Large Hadron Collider is 3.0%, what is the uncertainty in its position? How does this uncertainty affect the ability of scientists to collide such a proton with an atom (diameter roughly on the order of Angstroms, or 1x10-10 m) or another proton (charge radius about 0.9 fm) in the Large Hadron Collider? h ∆ x∙m∆u≥ 4π 6.626 x 10−34 Js h = ∆ x≥ 4 π ∙ m∆ u m 4 π ∙ 1.6726 x 10−27 kg ∙[ 0.030 x 0.999999991 × 2.998 × 108 ] s −15 ∆ x ≥ 3.5 x 10 m=3.5 fm Since the atom is more than 10,000 times larger than the size of the uncertainty in the proton’s position, colliding a proton with any part of the atom is not affected by this uncertainty (it is somewhat like hitting a car with a pea). However, the uncertainty is about twice the size of the diameter of a proton, so colliding 2 protons would be much more challenging.

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4. (a) What do wave functions (Ψ) tell you? Wavefunctions are solutions to Schrodinger’s wave equation for the matter wave of an electron in an atom. Wavefunctions define the allowed energy of an electron in a hydrogen atom; they do NOT describe the path of the electron’s motion. (b) What are quantum numbers and how are they related to the wave function? Each wavefunction (or solution to Schrodinger’s wave equation) is identified by a unique combination of three integers. These integers are called quantum numbers. (c) What does the square of the wave function (Ψ2) tell you? Ψ2 gives the probability of an electron with a certain energy existing anywhere inside the atom (d) What is an orbital? An orbital is the 3D region of space in an atom where the probability of finding an electron with that particular energy is high. 5. (a) 3.71 How does the concept of an orbit in the Bohr model of the hydrogen atom differ from the concept of an orbital in quantum theory? Orbits in the Bohr model describe distinct, concentric paths in which the electron moves as a particle around the nucleus. In contrast, an orbital is simply a region of space in which the probability of finding an electron of a certain energy is high, and the electron is assumed to move as a wave, not a particle.

(c) 3.73 How many quantum numbers are needed to identify an orbital? How many are needed to uniquely identify an electron in an atom? Why is there a difference? An orbital is related to a certain Ψ2. Since each wavefunction, Ψ, is identified by a combination of 3 unique integers (quantum numbers), each orbital is identified by those 3 unique quantum numbers. Since two electrons can exist in each orbital, 4 quantum numbers are needed to uniquely identify an electron in an atom; 3 quantum numbers identify the orbital of the electron, and the fourth number identifies the spin of the electron, which can be either +1/2 or -1/2.

6. For each of the 4 types of quantum numbers give the name of the quantum number, its letter/symbol, the range of its possible values, and explain in 1-3 sentences what property is related to each quantum number. a. The principle quantum number (n) can have any positive integer value, or n ≥ 1. The principle quantum number, n, corresponds to the energy level of the electron (shell) b. The angular momentum quantum number (ℓ) has n total values ranging from 0 to n – 1 in integer steps, or in other words ℓ = 0, 1, 2, … n-1. The angular momentum quantum number corresponds to the subshell which gives the shape of the orbital. c. The magnetic quantum number (ml) has 2ℓ +1 total values ranging from –ℓ to +ℓ in integer steps (including 0), or in other words ml = - ℓ … -2, -1, 0, 1, 2… ℓ . The magnetic quantum number corresponds to the specific orbital and gives the spatial orientation of that orbital. d. The spin quantum number ms has 2 possible values, + ½ and – ½. The spin quantum number gives the direction of the ‘spin’ of the electron in that orbital.

7. (a) 3.77 What are the possible values of quantum number l when n = 4? The angular momentum quantum number (ℓ) has n total values ranging from 0 to n – 1 in integer steps, so if n = 4, then ℓ can equal 0, 1, 2, 3. (b) 3.78 Which are the possible values of ml when l = 2? The magnetic quantum number (ml) has 2ℓ +1 total values ranging from –ℓ to +ℓ in integer steps (including 0), so if ℓ = 2, ml can equal -2, -1, 0, +1, +2. 8. (a) Which of the following sets of quantum numbers (n, l, ml, ms) are actually possible? a. 3, 2, -2, ½ b. 1, 1, 0, ½ ℓ can never equal n, only n-1 c. 1, 0, 1, -½ if ℓ=0, ml can only equal 0 d. 0, 0, 0, ½ n can never equal 0 e. 2, 0, 3, -½ if ℓ=0, ml can only equal 0...