Question

Among the following the wrong statement is (1) \Lambdas the number of orbit increases from the nucleus, the difference between the adjacent energy levels decreases. (2) The angular velocity of the ceectron in the \(n\) th orbit of Bohr's hydrogen atom is inversely proportional to \(n^{3}\). (3) According to Bohrs model of hydrogen atom, the angular momentum of the electron is quantised. (4) The total energy of an atomic electron is more than zero.

Short answer

Statement 4 is incorrect.

Step-by-step solution

Understanding the Problem

Determine which of the given statements is incorrect concerning the Bohr model of the hydrogen atom and its behavior.

Statement 1 Analysis

Check if the difference between adjacent energy levels decreases as the orbit's number increases. In Bohr's model, energy levels are given by \( E_n = -\frac{13.6}{n^2} \) eV. As \( n \) increases, the difference between adjacent energy levels gets smaller.

Statement 2 Analysis

Investigate the relationship between the angular velocity of the electron and the orbit number \( n \). Angular velocity \( \omega \) is given by \( \omega_n = \frac{2\pi k e^2 m}{n^3\hbar^3} \), which shows an inverse correlation with \( n^3 \).

Statement 3 Analysis

Validate if the angular momentum in Bohr's model is quantized. Angular momentum \( L \) is described as \( L = n\hbar \), indicating quantized values for different orbits.

Statement 4 Analysis

Check the total energy value of an atomic electron, which should be negative based on Bohr's model. Total energy is given by \( E_n = -\frac{13.6}{n^2} \) eV, always resulting in a value less than zero.

Key concepts

Energy Levels

In the Bohr model of the hydrogen atom, electrons occupy specific energy levels that are quantized. These energy levels are denoted by the principal quantum number, \(n\). The energy associated with each level is given by the formula: \[E_n = -\frac{13.6}{n^2} \text{ eV}\] As the principal quantum number \(n\) increases, the energy levels become closer to each other. This means that the difference in energy between consecutive levels decreases. Therefore, in statement (1), it is correctly noted that as the number of orbits increases, the difference between adjacent energy levels decreases. This understanding is crucial for explaining phenomena such as spectral lines and electronic transitions in atoms. Notice that energy levels are negative, which reflects that the electron is bound to the nucleus and energy must be supplied to free it.

Angular Momentum

Angular momentum in the Bohr model is an essential concept. According to Bohr, the angular momentum of an electron in a hydrogen atom is quantized and given by the equation: \[L = n\hbar\] Here, \(L\) represents angular momentum, \(n\) is the principal quantum number, and \(\hbar\) is the reduced Planck constant. This quantization means the electron can only occupy certain orbits with specific angular momentum values. Statement (3) correctly reflects this principle. The quantization of angular momentum leads to distinct energy levels and explains why electrons do not spiral into the nucleus, which classical mechanics would predict. Instead, they occupy stable orbits with quantized values of angular momentum.

Angular Velocity

Angular velocity describes how fast an electron orbits the nucleus in the Bohr model. It is given by the formula: \[\omega_n = \frac{2\pi k e^2 m}{n^3 \hbar^3}\] This equation shows that the angular velocity \(\omega\) of the electron is inversely proportional to the cube of the principal quantum number \(n\). Thus, as \(n\) increases, the angular velocity decreases rapidly. Statement (2) correctly states that angular velocity is inversely proportional to \(n^3\). Understanding angular velocity is important for grasping the behavior of electrons in different energy states and their speeds in those orbits.

Quantization

Quantization is a fundamental concept in the Bohr model and modern atomic theory. It refers to the idea that certain properties, like energy and angular momentum, can only take on discrete values. In the Bohr model, quantization explains why electrons have specific, fixed orbits and energy levels. For instance, the energy levels of a hydrogen atom are quantized so the electron can only exist in orbits with specific energy values. Angular momentum is also quantized, as described by: \ \[ L = n\hbar \] Mathess quantization prevents electrons from spiraling into the nucleus and instead leads to stable orbital patterns. Statement (4) is incorrect because, according to the Bohr model, the total energy of an atomic electron is always negative, indicating the electron is bound to the nucleus. Understanding quantization is crucial for explaining atomic stability and spectral lines observed in atomic emissions.