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MCQMediumJEE 2023Dual Nature of Matter & de Broglie Relation

JEE Chemistry 2023 Question with Solution

Electrons in a cathode ray tube have been emitted with a velocity of 1000 m s−11000 \, \text{m s}^{-1}. The number of following statements which is/are true about the emitted radiation is _____.

Given: h=6×10−34 J sh = 6 \times 10^{-34} \, \text{J s}, me=9×10−31 kgm_e = 9 \times 10^{-31} \, \text{kg}.

  • A

    The deBroglie wavelength of the electron emitted is 666.67 nm666.67 \, \text{nm}.

  • B

    The characteristic of electrons emitted depend upon the material of the electrodes of the cathode ray tube.

  • C

    The cathode rays start from cathode and move towards anode.

  • D

    The nature of the emitted electrons depends on the nature of the gas present in cathode ray tube.

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: Electrons are emitted with velocity v=1000 m s−1v = 1000 \, \text{m s}^{-1}, Planck constant h=6×10−34 J sh = 6 \times 10^{-34} \, \text{J s} and electron mass me=9×10−31 kgm_e = 9 \times 10^{-31} \, \text{kg}.

Find: The number of true statements.

For statement (A), use the de Broglie relation:

λ=hmv\lambda = \frac{h}{mv}

Substituting the given values:

λ=6×10−349×10−31×1000\lambda = \frac{6 \times 10^{-34}}{9 \times 10^{-31} \times 1000} λ=6.67×10−7 m=666.67 nm\lambda = 6.67 \times 10^{-7} \, \text{m} = 666.67 \, \text{nm}

So, statement (A) is true.

From the solution analysis, cathode rays are streams of electrons that start from the cathode and move towards the anode. Therefore, statement (C) is true.

The nature of electrons does not depend on the gas present in the cathode ray tube. The gas may affect fluorescence, but not the nature of the electrons. Therefore, statement (D) is false.

The solution is internally inconsistent for statement (B). One approach marks statement (B) true, while another states that the characteristic of emitted electrons is independent of electrode material. Since cathode rays are electrons and their fundamental nature is independent of electrode material, statement (B) is taken as false.

Hence, the true statements are (A) and (C) only. Therefore, the number of true statements is 22, and the correct option is B.

Statement-wise Evaluation

Given: v=1000 m s−1v = 1000 \, \text{m s}^{-1}, h=6×10−34 J sh = 6 \times 10^{-34} \, \text{J s}, me=9×10−31 kgm_e = 9 \times 10^{-31} \, \text{kg}.

Find: Count how many of the four statements are true.

  1. Statement (A):
λ=hmv=6×10−349×10−31×103\lambda = \frac{h}{mv} = \frac{6 \times 10^{-34}}{9 \times 10^{-31} \times 10^3} =6.67×10−7 m=666.67 nm= 6.67 \times 10^{-7} \, \text{m} = 666.67 \, \text{nm}

So this statement is true.

  1. Statement (B): Cathode rays are electrons. Their charge-to-mass behavior and basic nature do not depend on the electrode material. So this statement is false.
  1. Statement (C): Cathode rays originate from the cathode and move toward the anode. So this statement is true.
  1. Statement (D): The nature of electrons does not depend on the gas present in the tube. So this statement is false.

Thus, exactly 22 statements are true. Therefore, the correct option is B.

Common mistakes

  • Students often think statement (B) is true because electron emission may involve the cathode material. That is wrong here because the statement asks about the characteristic of electrons emitted, and the fundamental nature of electrons is independent of electrode material.

  • Many students confuse the nature of electrons with the effect of gas in the tube. The gas can affect fluorescence or discharge behavior, but it does not change the identity of the emitted electrons.

  • A common error is making a power-of-ten mistake while calculating de Broglie wavelength. Carefully substitute m=9×10−31 kgm = 9 \times 10^{-31} \, \text{kg} and v=103 m s−1v = 10^3 \, \text{m s}^{-1} before simplifying.

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