Step 1: Evaluate Statement (A)
Step 2: Evaluate Statement (B)
Step 3: Evaluate Statement (C)
According to Bohr’s model, the energy of an electron in the \( n \)-th orbit is given by:
\[ E = -\frac{13.6Z^2}{n^2} \text{ eV/atom}, \]
where \( Z \) is the atomic number.
Step 4: Evaluate Statement (D)
According to Bohr’s model, the velocity of an electron in the \( n \)-th orbit is given by:
\[ V = V_0 \frac{Z}{n}, \]
where \( V_0 \) is a constant.
Conclusion:
The correct statements are: (A, B, C).
To solve the problem, we need to evaluate each given statement about electrons in an atom and identify which are correct.
1. Uncertainty Principle and Electron Paths:
The Heisenberg Uncertainty Principle states that it is impossible to simultaneously know both the exact position and momentum of an electron.
Therefore, electrons do not have definite paths (or orbits), ruling out the classical idea of fixed electron paths.
Statement 1 is correct.
2. Energy of Electron in 2s Orbital vs. Electron at Infinite Distance:
The energy of an electron in any bound orbital (such as 2s) is negative and lower than the energy of a free electron at infinite distance (which is zero).
Thus, electron in 2s orbital has energy less than that of electron infinitely far.
Statement 2 is correct.
3. Most Negative Energy Value in Bohr’s Model:
Bohr’s model gives the energy levels as \(E_n = -\frac{13.6\, \text{eV}}{n^2}\). The lowest energy (most negative) corresponds to \(n=1\), which is the most stable orbit.
Statement 3 is correct.
4. Velocity of Electron and Quantum Number n in Bohr’s Model:
According to Bohr’s model, velocity \(v_n = \frac{Z e^2}{2 \varepsilon_0 h n}\), which means velocity decreases as \(n\) increases.
Therefore, velocity decreases with increasing \(n\).
Statement 4 is incorrect.
Final Answer:
Statements 1, 2, and 3 are correct; Statement 4 is incorrect.
Let $ a_0, a_1, ..., a_{23} $ be real numbers such that $$ \left(1 + \frac{2}{5}x \right)^{23} = \sum_{i=0}^{23} a_i x^i $$ for every real number $ x $. Let $ a_r $ be the largest among the numbers $ a_j $ for $ 0 \leq j \leq 23 $. Then the value of $ r $ is ________.
A temperature difference can generate e.m.f. in some materials. Let $ S $ be the e.m.f. produced per unit temperature difference between the ends of a wire, $ \sigma $ the electrical conductivity and $ \kappa $ the thermal conductivity of the material of the wire. Taking $ M, L, T, I $ and $ K $ as dimensions of mass, length, time, current and temperature, respectively, the dimensional formula of the quantity $ Z = \frac{S^2 \sigma}{\kappa} $ is: