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 m_l. 2. In Bohr's model, the kinetic energy E_k and the potential energy E_p of the hydrogen atom have the following relation: (a) E_k = 2E_p. (b) E_k = 1/2E_p. (c) E_k = -2E_p. (d) E_k = -1/2E_p. 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 4. The radial probability density function of the hydrogen atom is defined as: P_{nl}(r)dr = r^2R^2_{nl}(r)dr. For the 1s orbital, what is the relation between the averaged radius of the orbital <r> and the most probable radius of the orbital r_mp? The most probable radius of the orbital r_mp is defined as the radius at which one has the maximum probability of detecting the electron. (a) <r> > r_mp. (b) <r> = r_mp.
Added by Kyle B.
Close
Step 1
6 eV / n^2, where n is the principle quantum number. Show more…
Show all steps
Your feedback will help us improve your experience
Alkendra Singh and 71 other Organic Chemistry educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an $ orbital $, which may be thought of as a "cloud" of negative charge surrounding the nucleus. At the state of lowest energy, called the $ ground state $, or $ 1s-orbital $, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function $ p(r) = \frac{4}{a^3_0} r^2 e^{\frac{-2r}{a_0}} $ $ r \ge 0 $ where $ a_0 $ is the Bohr radius $ (a_0 \approx 5.59 \times 10^{-11} m) $. The integral $$ P(r) = \int_0^r \frac{4}{a^3_0} s^2 e^{\frac{-2r}{a_0}}\ ds $$ gives the probability that the electron will be found within the sphere of radius $ r $ meters centered at the nucleus. (a) Verify that $ p(r) $ is a probability density function. (b) Find $ lim_{r\to\infty} p(r) $. For what value of $ r $ does $ p(r) $ have its maximum value? (c) Graph the density function. (d) Find the probability that the electron will be within the sphere of radius $ 4_{a_0} $ centered at the nucleus. (e) Calculate the mean distance of the electron from the nucleus in the ground state of the hydrogen atom.
Further Applications of Integration
Probability
1) [1 mark] In the Bohr model of the hydrogen atom, an electron moves in a circular orbit around a proton. If the radius of this orbit is 5.3x10⁻¹¹ m and the speed of the electron is 2.2x10⁶ m/s what is the magnetic field generated at the center of the atom by the electrons motion? i) 10.2 T ii) 11.7 T iii) 12.5 T iv) 13.2 T 2) [1 mark] For the same Bohr hydrogen atom as described above, what is the electric current density associated with the electrons motion? Assume the atom is spherical. i) 5.6x10¹⁷ A/m² ii) 6.5x10¹⁷ A/m² iii) 7.6x10¹⁷ A/m² iv) 8.9x10¹⁷ A/m² 3) [1 mark] An electric point charge of +2.0 nC sits at the geometric center of a cube of side length 1.0 cm. What is the net electric flux through this cube? i) 110 Nm²/C ii) 138 Nm²/C iii) 195 Nm²/C iv) 226 Nm²/C
Bhushan A.
The hydrogen atom is composed of one proton in the nucleus and one electron, which moves about the nucleus. In the quantum theory of atomic structure, it is assumed that the electron does not move in a well-defined orbit. Instead, it occupies a state known as an orbital, which may be thought of as a "cloud" of negative charge surrounding the nucleus. At the state of lowest energy, called the ground state, or Is-orbital, the shape of this cloud is assumed to be a sphere centered at the nucleus. This sphere is described in terms of the probability density function $$p(r)=\frac{4}{a_{0}^{3}} r^{2} e^{-2 r / a_{0}} \quad r \geqslant 0$$ where $a_{0}$ is the Bohr radius$\left(a_{0} \approx 5.59 \times 10^{-11} \mathrm{m}\right) .$ The integral $$P(r)=\int_{0}^{r} \frac{4}{a_{0}^{3}} s^{2} e^{-2 s / a_{0}} d s$$ gives the probability that the electron will be found within the sphere of radius $r$ meters centered at the nucleus. \begin{equation} \begin{array}{l}{\text { (a) Verify that } p(r) \text { is a probability density function. }} \\ {\text { (b) Find } \lim _{r \rightarrow \infty} p(r) . \text { For what value of } r \text { does } p(r) \text { have its }} \\ {\text { maximum value? }}\\{\text { (c) Graph the density function. }} \\ {\text { (d) Find the probability that the electron will be within the }} \\ {\text { sphere of radius } 4 a_{0} \text { centered at the nucleus. }} \\ {\text { (e) Calculate the mean distance of the electron from the }} \\ {\text { nucleus in the ground state of the hydrogen atom. }}\end{array} \end{equation}
Recommended Textbooks
Organic Chemistry
Transcript
Watch the video solution with this free unlock.
EMAIL
PASSWORD
Organic Chemistry educators are ready to help you.