The de Broglie wavelength of an oxygen molecule at 27°C is x × 10⁻¹² m. The value of x is (take Planck's constant = 6.63 × 10⁻³⁴ J.s, Boltzmann constant = 1.38 × 10⁻²³ J/K, mass of oxygen. Molecule = 5.31 × 10⁻²⁶ kg).
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Question

The de Broglie wavelength of an oxygen molecule at 27°C is x × 10⁻¹² m. The value of x is (take Planck's constant = 6.63 × 10⁻³⁴ J.s, Boltzmann constant = 1.38 × 10⁻²³ J/K, mass of oxygen. Molecule = 5.31 × 10⁻²⁶ kg).

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G1 Physics · Accepted Answer

For a molecule in thermal motion, use the translational kinetic energy K = 3kT/2. The de Broglie wavelength is λ = h/p, with p = √(2mK). Therefore, λ = h/√(2mK) = h/√(3mkT). Here, T = 27 + 273 = 300 K. Substitute: h = 6.63 × 10⁻³⁴ J·s, m = 5.31 × 10⁻²⁶ kg, k = 1.38 × 10⁻²³ J/K, T = 300 K. Thus, λ = (6.63 × 10⁻³⁴)/ √[3×5.31×10⁻²⁶×1.38×10⁻²³×300] ≈ 2.58 × 10⁻¹¹ m = 25.8 × 10⁻¹² m. Hence, x ≈ 26. Correct option: A.