Problem 49
Question
According to Bohr's theory, \([2006-6 \mathrm{M}]\) \(E_{n}=\) Total energy, \(K_{n}=\) Kinetic energy, \(V_{n}=\) Potential energy, \(r_{n}=\) Radius of \(n^{\text {th }}\) orbit Match the following: Column I \(\quad\) Column II (A) \(V_{n} / K_{n}=\) ? (p) 0 (B) If radius of \(n^{\text {th }}\) orbit \(\propto E_{n}^{x},(q)-1\) \(x=?\) (C)Angular momentum in(r)-2 lowest orbital (D) (s) 1 \(\frac{1}{r_{n}} \propto Z^{y}, y=?\)
Step-by-Step Solution
Verified Answer
(A) matches (q), (B) and (C) unclear, (D) matches (s).
1Step 1: Explore Ratio of Potential to Kinetic Energy
According to Bohr's model, the potential energy \(V_n\) in a hydrogen-like atom is \( V_n = -2K_n \), where \(K_n\) is the kinetic energy. Therefore, \( \frac{V_n}{K_n} = -2 \). This matches with option \((q)\) -1.
2Step 2: Finding the Proportionality of Radius to Energy
In Bohr's model, the total energy \(E_n\) is inversely proportional to the radius \(r_n\) of the orbit, i.e., \( E_n \propto \frac{1}{r_n} \). Therefore, \( r_n \propto E_n^{-1} \). So, \(x = -1\). This matches with option \((q)\) -1.
3Step 3: Determining Angular Momentum in Lowest Orbital
In the lowest orbital (\(n=1\)), the angular momentum \(L\) is given by \( L = n \hbar = \hbar \). The relationship does not explicitly match any of the given options.
4Step 4: Proportionality of Inverse Radius to Atomic Number
According to Bohr's model, the radius \(r_n \propto 1/Z \), where \(Z\) is the atomic number. Thus, \( \frac{1}{r_n} \propto Z^{1} \), which means \(y = 1\). This matches with option \((s)\).
Key Concepts
Potential EnergyKinetic EnergyAngular MomentumAtomic Number
Potential Energy
In Bohr's model of the atom, potential energy in the context of electrons revolves around their position relative to the nucleus. Potential energy (V_n) of an electron in a hydrogen-like atom is negative, which denotes the attractive interaction between the negatively charged electron and the positively charged nucleus. This energy is calculated as a negative value because it requires energy input to move the electron out of its bound state.
Bohr's formula for potential energy is V_n = -2K_n, where K_n is the kinetic energy of the electron. This tells us that the potential energy is double and opposite in sign compared to the kinetic energy. By using the equation, the ratio \(\frac{V_n}{K_n} = -2\). Negative sign shows that the forces between electron and nucleus are attractive. For example, if the kinetic energy is a positive number, then potential energy will be twice its value but negative.
Bohr's formula for potential energy is V_n = -2K_n, where K_n is the kinetic energy of the electron. This tells us that the potential energy is double and opposite in sign compared to the kinetic energy. By using the equation, the ratio \(\frac{V_n}{K_n} = -2\). Negative sign shows that the forces between electron and nucleus are attractive. For example, if the kinetic energy is a positive number, then potential energy will be twice its value but negative.
Kinetic Energy
Kinetic energy (
K_n
) in Bohr's model refers to the motion of electrons orbiting the nucleus. It is the energy due to the electron’s velocity as it moves around the nucleus in quantized orbits. A higher kinetic energy means a faster electron orbit.
In Bohr's theory, the kinetic energy is linked with the attractive forces from the nucleus, allowing an electron to counterbalance the attractive electric forces and stabilize in orbit. This stabilization leads to quantized levels of energy, where only specific energy levels are permissible. The kinetic energy in an atom is important for understanding how electrons may jump to different energy levels, thus correlating with the concept of emission or absorption of energy as light.
The valuable insight from Bohr's model is that kinetic energy across various orbits accompanied by potential energy leads to stable electron configurations in atoms. These energy levels are quantized, key for explaining atomic spectra.
In Bohr's theory, the kinetic energy is linked with the attractive forces from the nucleus, allowing an electron to counterbalance the attractive electric forces and stabilize in orbit. This stabilization leads to quantized levels of energy, where only specific energy levels are permissible. The kinetic energy in an atom is important for understanding how electrons may jump to different energy levels, thus correlating with the concept of emission or absorption of energy as light.
The valuable insight from Bohr's model is that kinetic energy across various orbits accompanied by potential energy leads to stable electron configurations in atoms. These energy levels are quantized, key for explaining atomic spectra.
Angular Momentum
Angular momentum is a crucial quantity in Bohr's model of the atom. It represents the rotational equivalent of linear momentum, related to the motion of electrons in circular orbits around the nucleus.
In Bohr's model, angular momentum (L) is quantized. This means that electrons can only possess certain discrete values of angular momentum. The quantization is given by the formula: \(L = n\hbar\), where \(n\) is the principal quantum number (orbit number), and \(\hbar\) is the reduced Planck's constant.
For the lowest orbital with \(n = 1\), the angular momentum is simply \(\hbar\), representing the minimum amount of rotational energy an electron can have. This quantized aspect of angular momentum is fundamental to explaining why electrons do not spiral into the nucleus and why atoms emit light at specific wavelengths.
In Bohr's model, angular momentum (L) is quantized. This means that electrons can only possess certain discrete values of angular momentum. The quantization is given by the formula: \(L = n\hbar\), where \(n\) is the principal quantum number (orbit number), and \(\hbar\) is the reduced Planck's constant.
For the lowest orbital with \(n = 1\), the angular momentum is simply \(\hbar\), representing the minimum amount of rotational energy an electron can have. This quantized aspect of angular momentum is fundamental to explaining why electrons do not spiral into the nucleus and why atoms emit light at specific wavelengths.
Atomic Number
The atomic number (Z) is an essential concept in Bohr’s model as it defines the number of protons in an atom’s nucleus. It also dictates how many electrons are present in a neutral atom, thereby impacting the atom's size and energy levels.
In Bohr’s calculations, the atomic number plays a significant role in determining the radius of electron orbits. The radius of an electron’s orbit in Bohr’s model is inversely proportional to the atomic number, symbolized as \(r_n \propto \frac{1}{Z}\). This tells us that with an increase in atomic number, the radius decreases and thus the forces on electrons are stronger.
In Bohr’s calculations, the atomic number plays a significant role in determining the radius of electron orbits. The radius of an electron’s orbit in Bohr’s model is inversely proportional to the atomic number, symbolized as \(r_n \propto \frac{1}{Z}\). This tells us that with an increase in atomic number, the radius decreases and thus the forces on electrons are stronger.
- This relationship helps explain the variance in atomic sizes across elements.
- Higher atomic number means more protons, stronger pull on electrons, smaller orbitals.
Other exercises in this chapter
Problem 47
Isotopes of an element differ in the number of \(\ldots \ldots \ldots \ldots .\) in their nuclei.
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The energy of an electron in the first Bohr orbit of \(\mathrm{H}\) atom is \(-13.6 \mathrm{eV}\). The possible energy value(s) of the excited state(s) for elec
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The sum of the number of neutrons and proton in the isotope of hydrogen is : (a) 6 (b) 2 (c) 4 (d) 3
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