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Magazine Chapter 1: Problem 10
In an atom two electrons move around the nucleus in circular orbits or radii \(\mathrm{R}\) and \(4 \mathrm{R}\). The ratio of the time taken by them to complete one revolution is: (a) \(1: 4\) (b) \(4: 1\) (c) \(1: 8\) (d) \(8: 7\)
Short Answer
Expert verified
The ratio of the time taken is \(1:4\).
Step by step solution
01
Identify the Relevant Formula
The time taken for one complete revolution in a circular orbit is the circumference of the orbit divided by the orbital velocity of the electron. The formula for orbital velocity, in terms of orbital radius, is given by \( v = \frac{2\pi R}{T} \), where \( v \) is the velocity, \( R \) is the radius, and \( T \) is the time period for one complete revolution.
02
Express Orbital Velocity Differences
The centripetal force required for circular motion is provided by the electrostatic force of attraction between the nucleus and the electron. For two electrons in orbits of radii \( R \) and \( 4R \), the velocities \( v_1 \) and \( v_2 \) are different. The force relation can be expressed as \( \frac{kq^2}{R^2} = \frac{mv_1^2}{R} \) and \( \frac{kq^2}{(4R)^2} = \frac{mv_2^2}{4R} \) where \( k \) is Coulomb's constant, \( m \) is the electron mass, and \( q \) is the charge.
03
Derive the Relation Between Velocities
From the force equation, \( v_1^2 = \frac{kq^2}{mR} \) and \( v_2^2 = \frac{kq^2}{m(4R)} \). Solving these, \( v_2 = \frac{v_1}{2} \).
04
Calculate Time Period Ratio
Using \( T = \frac{2\pi R}{v} \), the time period \( T_1 \) for radius \( R \) is \( \frac{2\pi R}{v_1} \) and \( T_2 \) for radius \( 4R \) is \( \frac{8\pi R}{v_2} \). Substituting \( v_2 = \frac{v_1}{2} \), we get \( T_2 = \frac{8\pi R}{v_1/2} \).
05
Simplify the Result
This simplifies to \( T_2 = 4 \times \frac{2\pi R}{v_1} \) which is \( 4T_1 \). Therefore, the ratio of \( T_1 : T_2 \) is \( 1:4 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Electron Orbits
In an atom, electrons revolve around the nucleus in specific paths called orbits. These electron orbits are key to understanding atomic structure and behavior.
Each orbit is essentially a circular path in which an electron moves at a constant speed, determined by the electrostatic forces between the electron and nucleus.
This movement is analogous to planets orbiting the sun, governed by gravitational forces.
- An orbit's radius can affect the electron's speed and energy levels.
- In our exercise, the radii of two electron orbits are given as \(R\) and \(4R\), illustrating how an increase in orbit size impacts the electron's path.
Understanding electron orbits helps in comprehending not only atomic behavior but also chemical reactions and bonding.
Each orbit is essentially a circular path in which an electron moves at a constant speed, determined by the electrostatic forces between the electron and nucleus.
This movement is analogous to planets orbiting the sun, governed by gravitational forces.
- An orbit's radius can affect the electron's speed and energy levels.
- In our exercise, the radii of two electron orbits are given as \(R\) and \(4R\), illustrating how an increase in orbit size impacts the electron's path.
Understanding electron orbits helps in comprehending not only atomic behavior but also chemical reactions and bonding.
Orbital Velocity
Orbital velocity is the speed at which an electron moves around the nucleus. It determines how quickly an electron can complete a round trip along its orbit.
In the realm of atomic physics, the concept of orbital velocity connects to how energy levels are structured within the atom.
To find orbital velocity, we use the formula: \[ v = \frac{2\pi R}{T} \]Where \(v\) is the velocity, \(R\) is the radius of the orbit, and \(T\) is the time period for one complete revolution.
- A higher orbital velocity implies the electron is in a lower, more tightly-bound energy level.
- Conversely, a slower velocity indicates a higher energy level and a larger orbit.
Understanding orbital velocity is crucial for delving deep into phenomena such as electron configurations and spectroscopic transitions.
In the realm of atomic physics, the concept of orbital velocity connects to how energy levels are structured within the atom.
To find orbital velocity, we use the formula: \[ v = \frac{2\pi R}{T} \]Where \(v\) is the velocity, \(R\) is the radius of the orbit, and \(T\) is the time period for one complete revolution.
- A higher orbital velocity implies the electron is in a lower, more tightly-bound energy level.
- Conversely, a slower velocity indicates a higher energy level and a larger orbit.
Understanding orbital velocity is crucial for delving deep into phenomena such as electron configurations and spectroscopic transitions.
Coulomb's Law
Coulomb's law is fundamental in describing the forces acting on electrons in an atom.
It explains the electrostatic force of attraction or repulsion between charged particles, like electrons and protons.
Given by \[ F = \frac{kq_1q_2}{r^2} \]where:- \(F\) = force between the charges- \(k\) = Coulomb's constant- \(q_1\) and \(q_2\) = magnitudes of the charges- \(r\) = distance between the charges
- Within an atom, this force dictates the balance needed for circular motion.
This force provides the centripetal pull necessary for an electron to maintain its orbit.
Understanding how Coulomb's law applies within atoms helps unravel the interactions leading to atomic stability.
It explains the electrostatic force of attraction or repulsion between charged particles, like electrons and protons.
Given by \[ F = \frac{kq_1q_2}{r^2} \]where:- \(F\) = force between the charges- \(k\) = Coulomb's constant- \(q_1\) and \(q_2\) = magnitudes of the charges- \(r\) = distance between the charges
- Within an atom, this force dictates the balance needed for circular motion.
This force provides the centripetal pull necessary for an electron to maintain its orbit.
Understanding how Coulomb's law applies within atoms helps unravel the interactions leading to atomic stability.
Time Period Ratio
The time period ratio in this context refers to comparing the times it takes for electrons to complete an orbit.
For our example, it is crucial to understand how the size of the orbit affects this time period.
Using the relation: \[ T = \frac{2\pi R}{v} \]electrons in larger orbits, like \(4R\), take longer to complete a circle than those closer to the nucleus.
The exercise shows that for radii \(R\) and \(4R\), the corresponding times are \(T_1\) and \(T_2\).
- By solving the related formulas, we find that \(T_2 = 4T_1\) meaning the electron in a larger orbit takes four times as long to complete it.
This time period ratio helps students grasp how structural changes at the atomic level affect observable properties.
For our example, it is crucial to understand how the size of the orbit affects this time period.
Using the relation: \[ T = \frac{2\pi R}{v} \]electrons in larger orbits, like \(4R\), take longer to complete a circle than those closer to the nucleus.
The exercise shows that for radii \(R\) and \(4R\), the corresponding times are \(T_1\) and \(T_2\).
- By solving the related formulas, we find that \(T_2 = 4T_1\) meaning the electron in a larger orbit takes four times as long to complete it.
This time period ratio helps students grasp how structural changes at the atomic level affect observable properties.
