Q.49 The de Broglie wavelength of a proton moving at a speed of 1.0 m/s is _____ Å.
[Planck’s constant = 6.626 x 10-34 m2kg/s; mp = 1.67 x 10-27 kg]

De Broglie Wavelength of Proton at 1.0 m/s: Calculation and Explanation

The de Broglie wavelength of a proton moving at 1.0 m/s is 0.397 Å, calculated using the formula λ = h / (m v) with given values of Planck’s constant and proton mass.

Step-by-Step Calculation

The de Broglie wavelength λ relates to momentum p = m v via λ = h / p, where h is Planck’s constant.

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First, compute momentum: m = 1.67 × 10^{-27} kg, v = 1.0 m/s, so p = (1.67 × 10^{-27}) × 1.0 = 1.67 × 10^{-27} kg m/s.

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Then, λ = 6.626 × 10^{-34} / 1.67 × 10^{-27} = 3.968 × 10^{-7} m. Convert to Å (1 Å = 10^{-10} m): λ = 3.968 Å, rounded to 0.397 Å for typical precision in such problems.

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Formula Derivation

Louis de Broglie proposed that particles exhibit wave-particle duality, extending photon relations E = h ν and p = h / λ to matter waves.

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For non-relativistic speeds (v << c), p = m v, yielding λ = h / (m v). This holds here as 1.0 m/s is far below light speed.

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Common Options Analysis

In multiple-choice formats, options might include calculation errors:

• 3.97 Å: Correct but unrounded (3.968 ≈ 3.97).

• 39.7 Å: Error in unit conversion (forgot ×10^{-2} for Å).

• 0.0397 Å: Divided by 10 extra.

• 397 Å: Forgot 10^{-10} conversion factor entirely.

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The precise fill-in answer is 0.40 Å or 0.397 Å, emphasizing significant figures from given data.