In a multielectron atom, which of the following orbitals described by three quantum numbers with have same energy in absence of electric and magnetic fields? A. n = 1, l = 0, ml = 0$n = 1, l = 0, m_{l} = 0$ B. n = 2, l = 0, ml = 0$n = 2, l = 0, m_{l} = 0$ C. n = 2, l = 1, ml = 1$n = 2, l = 1, m_{l} = 1$ D. n = 3, l = 2, ml = 1$n = 3, l = 2, m_{l} = 1$ E. n = 3, l = 2, ml = 0$n = 3, l = 2, m_{l} = 0$ Choose the correct answer from the options given below:

Related Formula

Core Logic

For a constant principal quantum number n$n$, the energy E$E$ is directly proportional to -Z²$-Z^2$. This represents a quadratic relation where the curve is a downward-opening parabola starting from the origin in the negative energy region.

Pattern Recognition

Since energy values are inherently negative for bound states, as Z$Z$ increases, E$E$ becomes rapidly more negative following a parabolic curve.

Chapter Mix

Class 11 Chemistry: Structure of Atom

Related Formula

Core Logic

Statement I is a verbatim definition of Heisenberg's Uncertainty Principle, hence it is completely true.

For Statement II, we are given that Delta x = Delta p$Delta x = Delta p$:

Since Delta p = m cdot Delta v$Delta p = m cdot Delta v$:

This perfectly matches Statement II, so it is also true.

Pattern Recognition

When solving inequality bounds for identical uncertainties, always substitute directly to obtain a clean quadratic form before taking the square root.

Chapter Mix

Class 11 Chemistry: Structure of Atom

Related Formula

Core Logic

The ground-state electronic configuration of Chromium (Z=24$Z=24$) is anomalous due to half-filled stability:

Now count the electrons in specific subshells:

• For l = 1$l = 1$ (p-electrons): present in 2p⁶$2p^6$ and 3p⁶ 6 + 6 = 12$3p^6 \implies 6 + 6 = 12$ electrons.
• For l = 2$l = 2$ (d-electrons): present in 3d⁵ 5$3d^5 \implies 5$ electrons.

Thus, the counts are 12 and 5 respectively.

Pattern Recognition

Always remember the 3d⁵ 4s¹$3d^5 4s^1$ exception for Chromium (Z=24$Z=24$) and 3d¹⁰ 4s¹$3d^{10} 4s^1$ for Copper (Z=29$Z=29$). Forgetting the half-filled d-subshell exception will lead to the wrong answer (12 and 4).

Chapter Mix

Class 11 Chemistry: Structure of Atom

Core Logic

Let's evaluate each statement using quantum mechanics:

• Statement (1) is correct: The wave function squared Ψ²$\Psi^2$ (probability density) for a 1s orbital peaks directly at the nucleus (r = 0$r = 0$).
• Statement (2) is correct: The boundary distribution curves drop off exponentially but approach zero only at infinity, meaning the electron has a non-zero probability of being found at any distance, including 2a₀$2a_0$.
• Statement (3) is correct: All s-orbitals have an angular wave function component of unity, making them perfectly spherical.
• Statement (4) is incorrect: The total energy of a bound electron in a hydrogen atom is constant for a given principal quantum number quantum state (E = -13.6~eV$E = -13.6\mathrm{~eV}$ when n=1$n=1$), independent of its spatial distance from the nucleus.

Pattern Recognition

Total energy is a fixed state property defined strictly by the principal quantum number n$n$. It never varies as a function of the electron's position coordinates within that orbital state.

Chapter Mix

Class 11 Chemistry: Structure of Atom