Question

(a) Calculate the magnitude of the angular momentum for an l = 1 electron.

(b) Compare your answer to the value Bohr proposed for the n = 1 state.

Short answer

a) The magnitude of an object's angular momentum is\[1.492 \times {10^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}\]

b) The angular momentum in the part (a) is [1.414 times that of part (b).

Step-by-step solution

Define formula for angular momentum

The magnitude of angular momentum is given as

\[L = \left( {\sqrt {l\left( {l + 1} \right)} } \right)\left( {\frac{h}{{2\pi }}} \right)\]

Here,

L = Angular momentum

l = Angular momentum quantum

h = Planck's constant

Calculating the magnitude of the angular momentum

a)

Consider the valuel= 1.

Magnitude of angular momentum is calculated as:

\[\begin{array}{c}L = \left( {\sqrt {1\left( {1 + 1} \right)} } \right)\left( {\frac{{6.63 \times {{10}^{ - 34}}}}{{2\pi }}} \right)\;{\rm{J}} \cdot {\rm{s}}\\L = \sqrt 2 \left( {1.055 \times {{10}^{ - 34}}} \right)\;{\rm{J}} \cdot {\rm{s}}\\L = 1.492 \times {10^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}\end{array}\]

Therefore, the magnitude of angular momentum for an object is \[l = 1\]electron is \[1.492 \times {10^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}\]

Step 3: Comparing our answer to the value Bohr proposed

(b)

Given magnitude of angular momentum for an l= 1.electron is \[1.492 \times {10^{ - 34}}\;\;{\rm{J}} \cdot {\rm{s}}\]Formula Used:

The expression of the Bohr's modal for angular momentum is given as

\[L = \frac{{nh}}{{2\pi }}\]

Here,

L = Angular momentum

n = electron state

h = Planck'sConstant

Calculation:

For n = 1,

Bohr's modal for angular momentum is given is calculated as:

\[\begin{array}{c}L = \frac{{\left( 1 \right)\left( {6.63 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}} \right)}}{{2\pi }}\\L = 1.055 \times {10^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}\end{array}\]

Divide the results of the angular momentum to obtain the results as:

\[\frac{{1.492 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}}}{{1.055 \times {{10}^{ - 34}}\;{\rm{J}} \cdot {\rm{s}}}} = 1.414\]

Therefore, the angular momentum in the part (a) is 1.414 times that of part (b).