STATEMENT-1 : Band gap in germanium is small. STATEMENT-2 : The energy gap of each germanium atomic energy level is infinitesimally small.

Statement - 1 is True, Statement - 2 is False

Statement - 1 is True, Statement - 2 is True; Statement - 2 is a correct explanation for Statement - 1

Statement - 1 is True, Statement - 2 is True; Statement - 2 is not a correct explanation for Statement - 1

Statement - 1 is False, Statement - 2 is True

P Block Elements PYQ — JEE Advanced 2007 · Shift 2 · Q2 with Solution

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Correct answer: C

Definition of Band Gap: The band gap (or energy gap) in a solid is the energy difference between the top of the valence band and the bottom of the conduction band. Electrons must gain at least this amount of energy to be excited from the valence band to the conduction band, where they can conduct electricity.
Classification of Materials: Materials are classified based on their band gap size:
- Conductors (Metals): Have no band gap; the valence and conduction bands overlap. Electrons can move freely.
- Insulators: Have a large band gap (typically > 3 eV). A large amount of energy is needed to excite electrons, so they are poor conductors.
- Semiconductors: Have a small, non-zero band gap (typically between 0.1 and 3 eV). They can conduct electricity under certain conditions (e.g., by heating or doping).
Germanium (Ge) Properties: Germanium is a classic semiconductor. Its band gap is approximately $0.67$ eV at room temperature ( $300$ K). This value is small compared to insulators (e.g., diamond's band gap is $\sim 5.5$ eV), but larger than zero.
Conclusion for Statement-1: Because germanium's band gap allows for electrical conductivity under moderate conditions, it is considered small. Therefore, Statement-1 is True.

Atomic Energy Levels: In an isolated atom (like a single germanium atom), electrons occupy discrete, quantized energy levels (e.g., 1s, 2s, 2p, 3s, ...). These are called atomic energy levels or atomic orbitals.
Energy Gaps in an Atom: The energy differences between these discrete atomic levels are significant and are not infinitesimally small. For example, the energy required to excite an electron from one principal shell to the next is typically on the order of several electron volts (eV).
Formation of Energy Bands: When a large number of atoms come together to form a solid crystal, the Pauli exclusion principle dictates that the discrete atomic energy levels of the individual atoms must split and broaden into continuous bands of allowed energy levels. The energy levels within a single band are very closely spaced (quasi-continuous), meaning the gap between adjacent levels within the band is infinitesimally small. However, the statement refers to the gap of each atomic energy level, which is best interpreted as the gaps between the discrete levels of an isolated atom.
Conclusion for Statement-2: The energy gaps between the discrete atomic energy levels of a single germanium atom are large and well-defined, not infinitesimally small. The statement seems to confuse the properties of an isolated atom with the properties of energy bands in a solid. Therefore, Statement-2 is False.

This corresponds to option C.

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Correct answer: C

Definition of Band Gap: The band gap (or energy gap) in a solid is the energy difference between the top of the valence band and the bottom of the conduction band. Electrons must gain at least this amount of energy to be excited from the valence band to the conduction band, where they can conduct electricity.
Classification of Materials: Materials are classified based on their band gap size:
- Conductors (Metals): Have no band gap; the valence and conduction bands overlap. Electrons can move freely.
- Insulators: Have a large band gap (typically > 3 eV). A large amount of energy is needed to excite electrons, so they are poor conductors.
- Semiconductors: Have a small, non-zero band gap (typically between 0.1 and 3 eV). They can conduct electricity under certain conditions (e.g., by heating or doping).
Germanium (Ge) Properties: Germanium is a classic semiconductor. Its band gap is approximately $0.67$ eV at room temperature ( $300$ K). This value is small compared to insulators (e.g., diamond's band gap is $\sim 5.5$ eV), but larger than zero.
Conclusion for Statement-1: Because germanium's band gap allows for electrical conductivity under moderate conditions, it is considered small. Therefore, Statement-1 is True.

Atomic Energy Levels: In an isolated atom (like a single germanium atom), electrons occupy discrete, quantized energy levels (e.g., 1s, 2s, 2p, 3s, ...). These are called atomic energy levels or atomic orbitals.
Energy Gaps in an Atom: The energy differences between these discrete atomic levels are significant and are not infinitesimally small. For example, the energy required to excite an electron from one principal shell to the next is typically on the order of several electron volts (eV).
Formation of Energy Bands: When a large number of atoms come together to form a solid crystal, the Pauli exclusion principle dictates that the discrete atomic energy levels of the individual atoms must split and broaden into continuous bands of allowed energy levels. The energy levels within a single band are very closely spaced (quasi-continuous), meaning the gap between adjacent levels within the band is infinitesimally small. However, the statement refers to the gap of each atomic energy level, which is best interpreted as the gaps between the discrete levels of an isolated atom.
Conclusion for Statement-2: The energy gaps between the discrete atomic energy levels of a single germanium atom are large and well-defined, not infinitesimally small. The statement seems to confuse the properties of an isolated atom with the properties of energy bands in a solid. Therefore, Statement-2 is False.

This corresponds to option C.