Dual Nature of Matter: NEET MCQs with Answers

Explanation: First find the uncertainty in velocity from the stated accuracy: Δv = 0.001% of (3 × 10⁴) = (0.001/100) × 3 × 10⁴ = 0.3 cm s⁻¹. Then apply Δx = h/(4πmΔv). All quantities are in CGS units, so the answer emerges in cm. Denominator = 4 × 3.14 × (9.1 × 10⁻²⁸) × 0.3 = 3.429 × 10⁻²⁷. Δx = (6.626 × 10⁻²⁷)/(3.429 × 10⁻²⁷) = 1.93 cm. Striking result: knowing the electron’s speed to within 0.001% leaves its position uncertain by almost 2 cm — enormously larger than an atom.

Explanation: The de-Broglie relation can be written directly in terms of momentum: λ = h/p, so p = h/λ. p = (6.625 × 10⁻³⁴)/(10⁻¹⁷) = 6.625 × 10⁻³⁴ × 10¹⁷ = 6.625 × 10⁻¹⁷ kg m s⁻¹. Neither the mass nor the velocity is needed — momentum follows from the wavelength alone.

Explanation: Heisenberg’s uncertainty principle, Δx · Δp ≥ h/4π, contains NO reference to the mass or identity of the particle. Since the position of the helium atom is known to the same precision (1.0 mm) as that of the electron, the right-hand side h/4π is unchanged. Therefore the minimum uncertainty in momentum must also be the same: Δp = 5.0 × 10⁻²⁶ kg ms⁻¹. Trap: the heavier mass of helium tempts students to scale the answer, but momentum uncertainty depends only on the positional uncertainty.

Explanation: From Δx · Δp ≥ h/4π, the minimum uncertainty in position is Δx = h/(4πΔp). Taking π = 22/7: Δx = (6.62 × 10⁻³⁴ × 7)/(4 × 22 × 1 × 10⁻⁵) = (46.34 × 10⁻³⁴)/(88 × 10⁻⁵) = 5.265 × 10⁻³⁰ ≈ 5.27 × 10⁻³⁰ m.

Explanation: λ = h/mv, where all quantities must be in SI units. Convert the mass: m = 1 g = 1 × 10⁻³ kg. λ = (6.63 × 10⁻³⁴)/(10⁻³ × 100) = (6.63 × 10⁻³⁴)/(10⁻¹) = 6.63 × 10⁻³³ m. Common error: forgetting to convert grams to kilograms, which shifts the answer by three powers of ten.

Explanation: From Δx · mΔv = h/4π, we get Δv = h/(4πmΔx). Convert the distance: Δx = 0.1 Å = 0.1 × 10⁻¹⁰ m = 1 × 10⁻¹¹ m. Denominator = 4 × 3.14 × (9.11 × 10⁻³¹) × (1 × 10⁻¹¹) = 1.144 × 10⁻⁴⁰. Δv = (6.626 × 10⁻³⁴)/(1.144 × 10⁻⁴⁰) = 5.79 × 10⁶ ms⁻¹. Note how large this is — confining an electron to atomic dimensions makes its velocity almost completely indeterminate, which is the essence of the uncertainty principle.

Explanation: Heisenberg’s uncertainty principle: Δx · Δp = h/4π. Given the condition Δx = Δp, substituting gives (Δp)² = h/4π. Since Δp = mΔv, we get m²(Δv)² = h/4π ⇒ (Δv)² = h/(4πm²) ⇒ Δv = (1/2m)√(h/π). The factor of ½ emerges from taking the square root of the 4 in the denominator.

Explanation: Momentum and velocity are related by Δp = m × Δv, so Δv = Δp/m. Both quantities are already in the same (CGS) system of units, so no conversion is needed. Δv = (1 × 10⁻¹⁸ g cm s⁻¹)/(9 × 10⁻²⁸ g) = 0.111 × 10¹⁰ ≈ 1 × 10⁹ cm s⁻¹. The uncertainty principle itself is not needed here — only the definition of momentum.

Explanation: The de-Broglie relation is λ = h/mv. λ = (6.6 × 10⁻³⁴)/(0.66 × 100) = (6.6 × 10⁻³⁴)/66 = 1.0 × 10⁻³⁵ m. This vanishingly small wavelength is precisely why wave behaviour is never observed for macroscopic objects — it is far too small to be detected by any experiment.

Explanation: A hydrogen-like (one-electron) species has no shielding, so orbital energy depends only on n and the 2s and 2p orbitals are degenerate; this statement is therefore wrong (it is true only for multi-electron atoms). The de Broglie relation, ΔE·Δt ≥ h/4π, and the extra stability of half-filled and fully filled orbitals are correct.

Explanation: From mvr = nh/2π and λ = h/mv, 2πr = nλ. For n = 2, r = 52.9 × 2² = 211.6 pm, so λ = 2π(211.6)/2 = 211.6π pm.

Explanation: Convert the frequency to SI: ν = 1368 kHz = 1368 × 10³ s⁻¹ = 1.368 × 10⁶ s⁻¹. Using c = νλ, λ = c/ν: λ = (3.0 × 10⁸ m s⁻¹)/(1.368 × 10⁶ s⁻¹) = 219.3 m. A wavelength of a few hundred metres is exactly what is expected for the medium-wave radio band.

Explanation: An m value of −3 needs l ≥ 3, and since l ≤ n − 1, the lowest such shell has n = 4. In the nth Bohr orbit the circumference equals nλ, so 4 de Broglie wavelengths fit.

Explanation: The de Broglie wavelength is λ = h/p, so it is inversely proportional to momentum. Reducing the momentum to one-third makes the wavelength three times larger: 3 × 3.0 × 10⁻¹⁰ = 9.0 × 10⁻¹⁰ m.

Explanation: The principle applies to all moving bodies, but the product of the uncertainties varies inversely with mass. For a body of several hundred kilograms h/(4πm) is so tiny that it lies far below anything measurable. The practical limit on a motorbike's data is the measuring equipment, not the principle, so the principle matters only for microscopic particles such as electrons.