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Which particle has the smallest de Broglie wavelength? A an electron moving at 4 × 10³ m s⁻¹ B a proton moving at 4 × 10³ m s⁻¹ C an electron moving at 8 × 10⁵ m s⁻¹ D a proton moving at 8 × 10⁵ m s⁻¹ - AQA - A-Level Physics - Question 17 - 2022 - Paper 1

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Question 17

Which-particle-has-the-smallest-de-Broglie-wavelength?--A-an-electron-moving-at-4-×-10³-m-s⁻¹--B-a-proton-moving-at-4-×-10³-m-s⁻¹--C-an-electron-moving-at-8-×-10⁵-m-s⁻¹--D-a-proton-moving-at-8-×-10⁵-m-s⁻¹-AQA-A-Level Physics-Question 17-2022-Paper 1.png

Which particle has the smallest de Broglie wavelength? A an electron moving at 4 × 10³ m s⁻¹ B a proton moving at 4 × 10³ m s⁻¹ C an electron moving at 8 × 10⁵ m ... show full transcript

Worked Solution & Example Answer:Which particle has the smallest de Broglie wavelength? A an electron moving at 4 × 10³ m s⁻¹ B a proton moving at 4 × 10³ m s⁻¹ C an electron moving at 8 × 10⁵ m s⁻¹ D a proton moving at 8 × 10⁵ m s⁻¹ - AQA - A-Level Physics - Question 17 - 2022 - Paper 1

Step 1

Identify the concept of de Broglie wavelength

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Answer

The de Broglie wavelength au au is given by the formula: au=hp au = \frac{h}{p} where hh is Planck's constant and pp is the momentum of the particle. The momentum can be calculated as the product of the mass and velocity of the particle: p=mvp = mv.

Step 2

Calculate the de Broglie wavelength for each particle

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Answer

  1. Electron moving at 4 × 10³ m s⁻¹ (mass m≈9.11×10−31m \approx 9.11 \times 10^{-31} kg): p=m×v=9.11×10−31 kg×4×103 m s−1≈3.644×10−27 kg m s−1p = m \times v = 9.11 \times 10^{-31} \text{ kg} \times 4 \times 10^{3} \text{ m s}^{-1} \approx 3.644 \times 10^{-27} \text{ kg m s}^{-1} τ≈6.626×10−34 J s3.644×10−27 kg m s−1≈1.82×10−7 m\tau \approx \frac{6.626 \times 10^{-34} \text{ J s}}{3.644 \times 10^{-27} \text{ kg m s}^{-1}} \approx 1.82 \times 10^{-7} \text{ m}

  2. Proton moving at 4 × 10³ m s⁻¹ (mass m≈1.67×10−27m \approx 1.67 \times 10^{-27} kg): p=m×v=1.67×10−27 kg×4×103 m s−1≈6.68×10−24 kg m s−1p = m \times v = 1.67 \times 10^{-27} \text{ kg} \times 4 \times 10^{3} \text{ m s}^{-1} \approx 6.68 \times 10^{-24} \text{ kg m s}^{-1} τ≈6.626×10−34 J s6.68×10−24 kg m s−1≈9.92×10−11 m\tau \approx \frac{6.626 \times 10^{-34} \text{ J s}}{6.68 \times 10^{-24} \text{ kg m s}^{-1}} \approx 9.92 \times 10^{-11} \text{ m}

  3. Electron moving at 8 × 10⁵ m s⁻¹: p=9.11×10−31×8×105≈7.29×10−25 kg m s−1p = 9.11 \times 10^{-31} \times 8 \times 10^{5} \approx 7.29 \times 10^{-25} \text{ kg m s}^{-1} τ≈6.626×10−347.29×10−25≈9.07×10−10 m\tau \approx \frac{6.626 \times 10^{-34}}{7.29 \times 10^{-25}} \approx 9.07 \times 10^{-10} \text{ m}

  4. Proton moving at 8 × 10⁵ m s⁻¹: p=1.67×10−27×8×105≈1.34×10−21 kg m s−1p = 1.67 \times 10^{-27} \times 8 \times 10^{5} \approx 1.34 \times 10^{-21} \text{ kg m s}^{-1} τ≈6.626×10−341.34×10−21≈4.94×10−13 m\tau \approx \frac{6.626 \times 10^{-34}}{1.34 \times 10^{-21}} \approx 4.94 \times 10^{-13} \text{ m}

Step 3

Compare the calculated wavelengths

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Answer

The de Broglie wavelengths calculated are approximately:

  • Electron at 4 × 10³ m s⁻¹: 1.82×10−71.82 \times 10^{-7} m
  • Proton at 4 × 10³ m s⁻¹: 9.92×10−119.92 \times 10^{-11} m
  • Electron at 8 × 10⁵ m s⁻¹: 9.07×10−109.07 \times 10^{-10} m
  • Proton at 8 × 10⁵ m s⁻¹: 4.94×10−134.94 \times 10^{-13} m

Thus, the particle with the smallest de Broglie wavelength is D a proton moving at 8 × 10⁵ m s⁻¹.

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