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Chapter 5: Problem 588

The ratio of angular momentum of the electron in the first allowed orbit to that in the second allowed orbit of hydrogen atom is ...... \(\\{\mathrm{A}\\} \sqrt{2}\) \(\\{B\\} \sqrt{(1 / 2)}\) \(\\{\mathrm{C}\\}(1 / 2) \quad\\{\mathrm{D}\\} 2\)

Short Answer

Expert verified
The ratio of the angular momentum of the electron in the first allowed orbit to that in the second allowed orbit of the hydrogen atom is \(\frac{1}{2}\). Therefore, the correct answer is (C) \(\frac{1}{2}\).

Step by step solution

01

Write down the formula for angular momentum in an allowed orbit

The formula for the angular momentum (L) in the allowed orbits according to Bohr's quantization condition is given by: \( L = n\frac{h}{2\pi} \) Where \(n\) is the orbit number (integer), and \(h\) is Planck's constant.
02

Calculate the angular momentum for the first allowed orbit

Let's find the angular momentum (L1) for the first allowed orbit (n=1). \( L_1 = 1\frac{h}{2\pi} = \frac{h}{2\pi} \)
03

Calculate the angular momentum for the second allowed orbit

Now, let's find the angular momentum (L2) for the second allowed orbit (n=2). \( L_2 = 2\frac{h}{2\pi} = \frac{2h}{2\pi} \)
04

Find the ratio of the angular momenta

Let's find the ratio of L1 to L2. \( \textup{Ratio} (\frac{L_1}{L_2}) = \frac{\frac{h}{2\pi}}{\frac{2h}{2\pi}} \)
05

Simplify the ratio

Simplify the ratio by cancelling out the common terms. \( \frac{L_1}{L_2} = \frac{\frac{h}{2\pi}}{\frac{2h}{2\pi}} = \frac{1}{2} \)
06

Select the correct answer from the options given

The correct answer is the one that matches our calculated ratio. The ratio of the angular momentum of the electron in the first allowed orbit to that in the second allowed orbit of the hydrogen atom is \(\frac{1}{2}\). So, the correct answer is: (C) \(\frac{1}{2}\).

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Most popular questions from this chapter

Let I be the moment of inertia of a uniform square plate about an axis \(\mathrm{AB}\) that passes through its centre and is parallel to two of its sides \(\mathrm{CD}\) is a line in the plane of the plate that passes through the centre of the plate and makes an angle of \(\theta\) with \(\mathrm{AB}\). The moment of inertia of the plate about the axis \(\mathrm{CD}\) is then equal to.... \(\\{\mathrm{A}\\} \mathrm{I}\) \(\\{B\\} I \sin ^{2} \theta\) \(\\{C\\} I \cos ^{2} \theta\) \(\\{\mathrm{D}\\} I \cos ^{2}(\theta / 2)\)

Match list I with list II and select the correct answer $$ \begin{aligned} &\begin{array}{|l|l|} \hline \text { List-I } & \begin{array}{l} \text { List - II } \\ \text { System } \end{array} & \text { Moment of inertia } \\ \hline \text { (x) A ring about it axis } & \text { (1) }\left(\mathrm{MR}^{2} / 2\right) \\ \hline \text { (y) A uniform circular disc about it axis } & \text { (2) }(2 / 5) \mathrm{MR}^{2} \\ \hline \text { (z) A solid sphere about any diameter } & \text { (3) }(7 / 5) \mathrm{MR}^{2} \\ \hline \text { (w) A solid sphere about any tangent } & \text { (4) } \mathrm{MR}^{2} \\ \cline { 2 } & \text { (5) }(9 / 5) \mathrm{MR}^{2} \\ \hline \end{array}\\\ &\text { Select correct option }\\\ &\begin{array}{|l|l|l|l|l|} \hline \text { Option? } & \mathrm{X} & \mathrm{Y} & \mathrm{Z} & \mathrm{W} \\\ \hline\\{\mathrm{A}\\} & 2 & 1 & 3 & 4 \\ \hline\\{\mathrm{B}\\} & 4 & 3 & 2 & 5 \\ \hline\\{\mathrm{C}\\} & 1 & 5 & 4 & 3 \\ \hline\\{\mathrm{D}\\} & 4 & 1 & 2 & 3 \\ \hline \end{array} \end{aligned} $$

If the earth were to suddenly contract so that its radius become half of it present radius, without any change in its mass, the duration of the new day will be... \(\\{\mathrm{A}\\} 6 \mathrm{hr}\) \\{B \(12 \mathrm{hr}\) \(\\{\mathrm{C}\\} 18 \mathrm{hr}\) \\{D \(\\} 30 \mathrm{hr}\)

Statement \(-1-\) The angular momentum of a particle moving in a circular orbit with a constant speed remains conserved about any point on the circumference of the circle. Statement \(-2\) - If no net torque outs, the angular momentum of a system is conserved. \(\\{\mathrm{A}\\}\) Statement \(-1\) is correct (true), Statement \(-2\) is true and Statement- 2 is correct explanation for Statement \(-1\) \\{B \\} Statement \(-1\) is true, statement \(-2\) is true but statement- 2 is not the correct explanation four statement \(-1\). \(\\{\mathrm{C}\\}\) Statement \(-1\) is true, statement- 2 is false \\{D \(\\}\) Statement- 2 is false, statement \(-2\) is true

A Pulley of radius \(2 \mathrm{~m}\) is rotated about its axis by a force \(F=\left(20 t-5 t^{2}\right) N\) where \(t\) is in sec applied tangentially. If the moment of inertia of the Pulley about its axis of rotation is $10 \mathrm{KgM}^{2}$, the number of rotations made by the pulley before its direction of motion is reversed is : \(\\{\mathrm{A}\\}\) more than 3 but less then 6 \(\\{\mathrm{B}\\}\) more than 6 but less then 9 \(\\{\mathrm{C}\\}\) more than 9 \\{D \\} Less then 3
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