Question 1

Consider a one-dimensional particle in a box of length 2L and one of length 3L.Which of the following is true?

Accepted Answer

The energy levels for a one-dimensional particle in a box are given by the equation:

$E_n=\frac{n^2h^2}{8mL^2}$

where n is the quantum number, h is Planck's constant, m is the mass of the particle, and L is the length of the box.

For the box of length 2L, the energy levels for n = 1 and n = 2 are:

$E_1=\frac{h^2}{8mL^2}$

and

$E_2=\frac{h^2}{2mL^2}$

Thus, the energy difference between the two levels is:

$E_2-E_1=\frac{3h^2}{8mL^2}$

For the box of length 3L, the energy levels for n = 1 and n = 2 are:

$E_1=\frac{h^2}{18mL^2}$

and

$E_2=\frac{h^2}{2mL^2}$

Thus, the energy difference between the two levels is:

$E_2-E_1=\frac{8h^2}{18mL^2}-\frac{h^2}{18mL^2}=\frac{7h^2}{18mL^2}$

Since $\frac{3}{2}<\frac{7}{3}$, the energy levels n = 1 and n = 2 are closer together for the box of length 3L. Therefore, the correct answer is A.

Note that the zero-point energy, which is the energy of the ground state (n=1), is:

$E_1=\frac{h^2}{8mL^2}$

and is the same for both boxes. Therefore, D is also a correct statement, but it is not the best choice because it does not address the energy levels for n=2. Choice C is not correct, as it implies that all energy levels are shifted by the same amount for the box of length 3L, which is not true. Choice B and E are also not correct, as they do not account for the correct ordering of the energy levels.

Question 2

Which of the following metals has the lowest work function?

Accepted Answer

Rb (Rubidium) has the lowest work function among the options given. Work function is the minimum energy needed to remove an electron from the surface of a metal. In general, alkali metals like Rb have lower work functions due to their single valence electron, which is loosely bound compared to elements in other groups.

Question 3

The zero-point energy in 1,3-butadiene is larger than in 1,3,5,7-octatetraene.True or false?

Accepted Answer

The zero-point energy is directly proportional to the number of double bonds in a molecule. 1,3-butadiene has two double bonds, while 1,3,5,7-octatetraene has four, so the zero-point energy in 1,3-butadiene is larger.

Question 4

Calculate the longest-wavelength line in the Balmer series for hydrogen.

Accepted Answer

The Balmer series for hydrogen is given by the formula 1/λ = R(1/22 - 1/n2), where R is the Rydberg constant and n is an integer representing the energy level of the electron. The longest-wavelength line would be when n=3 and the formula becomes 1/λ = R(1/22 - 1/32). Solving for λ gives a value of 656.3 nm, which is closest to answer choice B.

Question 5

Comparing the lowest energy transition,$\Delta$E,in1,3-butadiene(A)and 1,3,5,7-octate$\Delta$traene(B),$\Delta$E_A is greater than $\Delta$E_B.True or false?

Accepted Answer

The energy of lowest transition in 1,3-butadiene is F1F1F1S1F1F1F10E, while in 1,3,5,7-octatetraene it is F1F1F1S1F1F1F10E_A. Since the latter involves an additional excited state and has more complex electronic transitions, it has a higher energy than the former. Therefore, F1F1F1S1F1F1F10E_A is greater than F1F1F1S1F1F1F10E_B in terms of energy of lowest transition.

Question 6

The Balmer series of lines for the hydrogen atom are found only in the visible region of the spectrum.True or false?

Accepted Answer

The Balmer series primarily falls within the visible spectrum, but it also includes lines in the ultraviolet region.

Question 7

What is the probability of finding an electron in a small region of a hydrogen 1s orbital at a distance 3a₀ from the nucleus relative to the probability of finding it in the same small region located at the nucleus?

Accepted Answer

The answer of What is the probability of finding an...

Question 8

The four  $\pi$ electrons of 1,3-butadiene can be modeled as particles in a box.If the length of 1,3-butadiene (L)is 0.56 nm,calculate the energy of the lowest energy transition.

Accepted Answer

The answer of The four  $\pi$ electrons of 1,3-butadiene...

Question 9

Which of the following is expected to approximate the value of the wavelength of a 5-g bullet traveling at 1300 miles per hour?

Accepted Answer

The answer of Which of the following is expected to...

Question 10

The planet Mercury has a maximum intensity of solar radiation of 3700 nm on the sunlit side.What is the temperature of the surface of Mercury on the sunlit side? The empirical value of the constant in Wien's law is 2.9 K·mm.

Accepted Answer

The answer of The planet Mercury has a maximum intensity...

Question 11

Which of the following statements with respect to the photoelectric effect is incorrect?

Accepted Answer

The slope of the plot of kinetic energy of the ejected electrons versus the frequency of the incident radiation is equal to Planck's constant (h), not the value of the work function. The work function is the y-intercept of this plot, representing the minimum energy needed to eject electrons.

Question 12

All the orbitals of a given shell of He⁺ are degenerate regardless of the value of the orbital angular momentum.True or false?

Accepted Answer

The orbitals of a given shell of an atom are always degenerate regardless of the value of the orbital angular momentum. This is because they share the same principal quantum number (n) and therefore have the same energy level.

Question 13

What is the shortest-wavelength line in the emission spectrum of the hydrogen atom?

Accepted Answer

The shortest-wavelength line in the emission spectrum of the hydrogen atom corresponds to the transition from the n=2 energy level to the n=1 energy level. This is known as the Lyman series, and the shortest line in this series has a wavelength of 91.2 nm. Therefore, the correct choice is (D).

Question 14

Which of the following is Wien's law?

Accepted Answer

Wien's law states that the wavelength at which the intensity of black-body radiation is maximum is inversely proportional to the temperature, which is most closely represented by choice A.

Question 15

The square of the amplitude of a wave determines the intensity of the radiation.True or false?

Accepted Answer

The intensity of radiation is directly proportional to the square of the amplitude of the wave.

Question 16

Carotene,which is partly responsible for the color of carrots,has 22  $\pi$ electrons in a conjugated system and a length of about 3 nm.The lowest energy transition in carotene is from n = ___ to n = ___.

Accepted Answer

The answer of Carotene,which is partly responsible for the color...

Question 17

Estimate the minimum uncertainty in the position of an electron of mass 9.109 *10 ^- ³¹ kg,given that its speed is unknown to within ±3.00 *10⁵m·s ^- ¹.

Accepted Answer

We can use the Heisenberg uncertainty principle, which states that the product of the uncertainties in position and momentum of a particle is at least equal to Planck's constant divided by 4*pi. Mathematically, Δx*Δp >= h/(4π). We are given the uncertainty in speed, not momentum, but we know that momentum is equal to mass times velocity. Therefore, we can rewrite the uncertainty in speed as an uncertainty in momentum: Δp = mΔv. Plugging this into the uncertainty principle, we get Δx*Δ(mΔv) >= h/(4π). Solving for Δx, we get Δx >= h/(4πΔ(mΔv)). Plugging in the given values and solving, we get Δx = h/(4πΔ(mΔv)) = (6.626 x 10^-34 J*s)/(4π(9.109 x 10^-31 kg)(3.00 x 10^5 m/s)(±3.00 x 10^15 m/s)) = 193 pm. Therefore, the minimum uncertainty in position is 193 pm, which corresponds to choice C.

Question 18

A node is a point at which the wavefunction becomes zero.True or false?

Accepted Answer

The answer of A node is a point at which...

Question 19

Which of the following emission lines corresponds to part of the Balmer series of lines in the spectrum of a hydrogen atom? (a)n₂ $\rightarrow$ n₁ (b)n₄ $\rightarrow$ n₂ (c)n₄ $\rightarrow$ n₁ (d)n₃ $\rightarrow$ n₂ (e)n₄ $\rightarrow$ n₃

Accepted Answer

The answer of Which of the following emission lines corresponds...

Quiz 1: The Quantum World

Consider a one-dimensional particle in a box of length 2L and one of length 3L.Which of the following is true?

Which of the following metals has the lowest work function?

The zero-point energy in 1,3-butadiene is larger than in 1,3,5,7-octatetraene.True or false?

Calculate the longest-wavelength line in the Balmer series for hydrogen.

Comparing the lowest energy transition, $\Delta$ E,in1,3-butadiene(A)and 1,3,5,7-octate $\Delta$ traene(B), $\Delta$ E_A is greater than $\Delta$ E_B.True or false?

The Balmer series of lines for the hydrogen atom are found only in the visible region of the spectrum.True or false?

What is the probability of finding an electron in a small region of a hydrogen 1s orbital at a distance 3a₀ from the nucleus relative to the probability of finding it in the same small region located at the nucleus?

The four $\pi$ electrons of 1,3-butadiene can be modeled as particles in a box.If the length of 1,3-butadiene (L)is 0.56 nm,calculate the energy of the lowest energy transition.

Which of the following is expected to approximate the value of the wavelength of a 5-g bullet traveling at 1300 miles per hour?

The planet Mercury has a maximum intensity of solar radiation of 3700 nm on the sunlit side.What is the temperature of the surface of Mercury on the sunlit side? The empirical value of the constant in Wien's law is 2.9 K·mm.

Which of the following statements with respect to the photoelectric effect is incorrect?

All the orbitals of a given shell of He⁺ are degenerate regardless of the value of the orbital angular momentum.True or false?

What is the shortest-wavelength line in the emission spectrum of the hydrogen atom?

Which of the following is Wien's law?

The square of the amplitude of a wave determines the intensity of the radiation.True or false?

Carotene,which is partly responsible for the color of carrots,has 22 $\pi$ electrons in a conjugated system and a length of about 3 nm.The lowest energy transition in carotene is from n = _ to n = _.

Estimate the minimum uncertainty in the position of an electron of mass 9.109 10 ^- ³¹ kg,given that its speed is unknown to within ±3.00 10⁵m·s ^- ¹.

A node is a point at which the wavefunction becomes zero.True or false?

Which of the following emission lines corresponds to part of the Balmer series of lines in the spectrum of a hydrogen atom? (a) n₂ $\rightarrow$ n₁ (b) n₄ $\rightarrow$ n₂ (c) n₄ $\rightarrow$ n₁ (d) n₃ $\rightarrow$ n₂ (e) n₄ $\rightarrow$ n₃

Quiz 1: The Quantum World

Consider a one-dimensional particle in a box of length 2L and one of length 3L.Which of the following is true?

Which of the following metals has the lowest work function?

The zero-point energy in 1,3-butadiene is larger than in 1,3,5,7-octatetraene.True or false?

Calculate the longest-wavelength line in the Balmer series for hydrogen.

Comparing the lowest energy transition, Δ\DeltaΔ E,in1,3-butadiene(A)and 1,3,5,7-octate Δ\DeltaΔ traene(B), Δ\DeltaΔ EA is greater than Δ\DeltaΔ EB.True or false?

The Balmer series of lines for the hydrogen atom are found only in the visible region of the spectrum.True or false?

What is the probability of finding an electron in a small region of a hydrogen 1s orbital at a distance 3a0 from the nucleus relative to the probability of finding it in the same small region located at the nucleus?

The four π\piπ electrons of 1,3-butadiene can be modeled as particles in a box.If the length of 1,3-butadiene (L)is 0.56 nm,calculate the energy of the lowest energy transition.

Which of the following is expected to approximate the value of the wavelength of a 5-g bullet traveling at 1300 miles per hour?

The planet Mercury has a maximum intensity of solar radiation of 3700 nm on the sunlit side.What is the temperature of the surface of Mercury on the sunlit side? The empirical value of the constant in Wien's law is 2.9 K·mm.

Which of the following statements with respect to the photoelectric effect is incorrect?

All the orbitals of a given shell of He+ are degenerate regardless of the value of the orbital angular momentum.True or false?

What is the shortest-wavelength line in the emission spectrum of the hydrogen atom?

Which of the following is Wien's law?

The square of the amplitude of a wave determines the intensity of the radiation.True or false?

Carotene,which is partly responsible for the color of carrots,has 22 π\piπ electrons in a conjugated system and a length of about 3 nm.The lowest energy transition in carotene is from n = ___ to n = ___.

Estimate the minimum uncertainty in the position of an electron of mass 9.109 *10 - 31 kg,given that its speed is unknown to within ±3.00 *105m·s - 1.

A node is a point at which the wavefunction becomes zero.True or false?

Which of the following emission lines corresponds to part of the Balmer series of lines in the spectrum of a hydrogen atom? (a) n2 →\rightarrow→ n1 (b) n4 →\rightarrow→ n2 (c) n4 →\rightarrow→ n1 (d) n3 →\rightarrow→ n2 (e) n4 →\rightarrow→ n3

Comparing the lowest energy transition, $\Delta$ E,in1,3-butadiene(A)and 1,3,5,7-octate $\Delta$ traene(B), $\Delta$ E_A is greater than $\Delta$ E_B.True or false?

What is the probability of finding an electron in a small region of a hydrogen 1s orbital at a distance 3a₀ from the nucleus relative to the probability of finding it in the same small region located at the nucleus?

The four $\pi$ electrons of 1,3-butadiene can be modeled as particles in a box.If the length of 1,3-butadiene (L)is 0.56 nm,calculate the energy of the lowest energy transition.

All the orbitals of a given shell of He⁺ are degenerate regardless of the value of the orbital angular momentum.True or false?

Carotene,which is partly responsible for the color of carrots,has 22 $\pi$ electrons in a conjugated system and a length of about 3 nm.The lowest energy transition in carotene is from n = _ to n = _.

Estimate the minimum uncertainty in the position of an electron of mass 9.109 10 ^- ³¹ kg,given that its speed is unknown to within ±3.00 10⁵m·s ^- ¹.

Which of the following emission lines corresponds to part of the Balmer series of lines in the spectrum of a hydrogen atom? (a) n₂ $\rightarrow$ n₁ (b) n₄ $\rightarrow$ n₂ (c) n₄ $\rightarrow$ n₁ (d) n₃ $\rightarrow$ n₂ (e) n₄ $\rightarrow$ n₃