Chemistry Q122 – AP-EAMCET 2025

Home Chemistry Dual Nature of MatterQuestion #380

If the kinetic energy of a particle having wavelength

x

Å is increased to three times, its de Broglie wavelength (in Å) is:

Clue 1 of 3: De Broglie Wavelength and Kinetic Energy

The de Broglie wavelength of a particle is inversely proportional to its momentum. Kinetic energy is related to momentum.

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Step 1: Relate de Broglie Wavelength to Kinetic Energy✦ Active

The de Broglie wavelength ( $λ$ ) is given by $λ = \frac{h}{p}$ , where $h$ is Planck's constant and $p$ is the momentum of the particle.

The kinetic energy ( $K E$ ) of a particle is related to its momentum by $K E = \frac{p^{2}}{2 m}$ , where $m$ is the mass of the particle.

From the kinetic energy formula, we can express momentum as $p = \sqrt{2 m K E}$ .

Substitute this expression for $p$ into the de Broglie wavelength formula:

λ = \frac{h}{\sqrt{2 m K E}}

💡 Teacher's Secret Hint

Remember that $h$ and $m$ are constants for a given particle, so $λ$ is inversely proportional to the square root of $K E$ .

Step 2: Set up Initial Conditions○ Expand

Let the initial de Broglie wavelength be $λ_{1} = x$ .

Let the initial kinetic energy be $K E_{1}$ .

So, for the initial state:

λ_{1} = x = \frac{h}{\sqrt{2 m K E_{1}}}

Step 3: Set up Final Conditions○ Expand

The kinetic energy is increased to three times its initial value, so the new kinetic energy is $K E_{2} = 3 K E_{1}$ .

Let the new de Broglie wavelength be $λ_{2}$ .

For the final state:

λ_{2} = \frac{h}{\sqrt{2 m K E_{2}}}

Step 4: Calculate the New Wavelength○ Expand

Substitute $K E_{2} = 3 K E_{1}$ into the equation for $λ_{2}$ :

λ_{2} = \frac{h}{\sqrt{2 m (3 K E_{1})}}

λ_{2} = \frac{h}{\sqrt{3 \cdot (2 m K E_{1})}}

λ_{2} = \frac{1}{\sqrt{3}} \cdot \frac{h}{\sqrt{2 m K E_{1}}}

From Step 2, we know that $\frac{h}{\sqrt{2 m K E_{1}}} = λ_{1} = x$ .

Therefore, the new de Broglie wavelength is:

λ_{2} = \frac{x}{\sqrt{3}}

This corresponds to option 3.

💡 Teacher's Secret Hint

Be careful with the square root. An increase in kinetic energy leads to a decrease in wavelength, so the new wavelength should be smaller than the original.

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Chemistry Question 122 – AP-EAMCET 2025

Clue 1 of 3: De Broglie Wavelength and Kinetic Energy

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