Question

using bohr atomic model , derive expression for calculating the radius of orbits in He+ . using this expression , calculate the radius of fourth orbit of He+ ion .

Answer 1

Answer on Question #66797 - Chemistry | Inorganic Chemistry

Using Bohr atomic model, derive expression for calculating the radius of orbits in $\mathrm{He^{+}}$ . Using this expression, calculate the radius of fourth orbit of $\mathrm{He^{+}}$ ion

Solution:

1. Derivation of expression for calculating the radius of orbits in $\mathrm{He^{+}}$ . Bohr atomic model:

$$F _ {\text {c e n t r i f u g a l}} = - m v ^ {2} / r$$ $$F _ {c o u l o m b i c} = - Z e ^ {2} / r ^ {2}$$ $$\frac {m v ^ {2}}{r} = \frac {Z e ^ {2}}{r ^ {2}} \Rightarrow r = \frac {m v ^ {2} r ^ {2}}{Z e ^ {2}}$$ $$r = \frac {m v ^ {2} r ^ {2}}{Z e ^ {2}} \times \frac {m}{m} = \frac {m ^ {2} v ^ {2} r ^ {2}}{m Z e ^ {2}} = \frac {(m v r) ^ {2}}{m Z e ^ {2}}$$

Quantum hypothesis:

$$m v r = n h / 2 \pi$$

So:

$$r = \frac {n ^ {2} h ^ {2}}{4 \pi^ {2} m Z e ^ {2}}$$

For the Hydrogen atom $(Z = 1)$ , the smallest radius $(n = 1)$ will be:

$$a _ {0} = \frac {1 * h ^ {2}}{4 \pi^ {2} m * 1 * e ^ {2}} = 0. 5 2 9 \AA$$

$a_0 - \text{Bohr radius, constant}$

So:

$$r = \frac {n ^ {2} a _ {0}}{Z}$$

For $\mathrm{He^{+}}$ ion $(Z = 2)$ , the calculation of radius is:

$$r = \frac {n ^ {2} a _ {0}}{2}$$

2. Calculation of the radius of fourth orbit of $\mathrm{He^{+}}$ ion.

$$n = 4$$ $$r = \frac {4 ^ {2} \times 0 . 5 2 9 \AA}{2} = 4. 2 3 2 \AA$$

Answer: The radius of fourth orbit of $\mathrm{He^{+}}$ ion is 4.232 Å.

Answer provided by http://www.AssignmentExpert.com/