Problem 8 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

The de Broglie wavelength of a car of mass \(1000 \mathrm{~kg}\) and velocity \(36 \mathrm{~km} / \mathrm{hr}\) is : [Main Online April 23, 2013] (a) \(6.626 \times 10^{-34} \mathrm{~m}\) (b) \(6.626 \times 10^{-38} \mathrm{~m}\) (c) \(6.626 \times 10^{-31} \mathrm{~m}\) (d) \(6.626 \times 10^{-30} \mathrm{~m}\)

Step-by-Step Solution

Answer

The de Broglie wavelength of the car is \(6.626 \times 10^{-38} \text{ m}\). Option (b) is correct.

Step 1: Convert Velocity to Meters per Second

The velocity of the car is given as \( 36 \text{ km/hr} \). To use it in the de Broglie wavelength formula, we first need to convert it to meters per second (m/s). We know that \( 1 \text{ km} = 1000 \text{ m} \) and \( 1 \text{ hour} = 3600 \text{ seconds} \). Therefore, the conversion is done as follows:\[36 \text{ km/hr} = \frac{36 \times 1000}{3600} \text{ m/s} = 10 \text{ m/s} \]

Step 2: Understand de Broglie Equation

The de Broglie wavelength \( \lambda \) is given by the formula:\[\lambda = \frac{h}{mv} \]where \( h \) is Planck's constant \( 6.626 \times 10^{-34} \text{ Js} \), \( m \) is the mass, and \( v \) is the velocity.

Step 3: Substitute Values into the Equation

The mass of the car \( m \) is \( 1000 \text{ kg} \) and the velocity \( v \) is \( 10 \text{ m/s} \). Substitute these values into the de Broglie equation:\[\lambda = \frac{6.626 \times 10^{-34} \text{ Js}}{1000 \text{ kg} \times 10 \text{ m/s}} \]

Step 4: Calculate de Broglie Wavelength

Calculate the wavelength by performing the division:\[\lambda = \frac{6.626 \times 10^{-34}}{10000} = 6.626 \times 10^{-38} \text{ m} \]

Step 5: Identify Correct Answer Choice

The calculated de Broglie wavelength \( 6.626 \times 10^{-38} \text{ m} \) corresponds to option (b).

Key Concepts

Planck's Constant

Planck's constant is a fundamental constant in physics, denoted by the symbol \( h \). This constant has a value of \( 6.626 \times 10^{-34} \text{ Js} \). It is crucial in the realm of quantum mechanics because it provides a relationship between the energy of a photon and its frequency. This relationship is expressed through the equation: \[ E = h \cdot f \]where \( E \) is energy and \( f \) is the frequency of the photon. When it comes to the de Broglie wavelength, Planck's constant is involved in the calculation of the wavelength associated with moving particles, such as electrons or larger objects like cars. This concept shows that all matter exhibits wave-like properties, especially at microscopic levels, and is central to understanding wave-particle duality. In the de Broglie equation:\[ \lambda = \frac{h}{mv} \]Planck's constant \( h \) acts as a scaling factor that is critical for converting the product of mass \( m \) and velocity \( v \) into a wavelength \( \lambda \). The small value of \( h \) reflects the very tiny wavelengths that matter can have when at rest or moving with relatively low velocities.

Mass Velocity Relationship

The mass and velocity of an object are key parameters in determining its de Broglie wavelength. Understanding how these two factors interact is crucial for applying the de Broglie equation. 1. **Mass \( m \):** - The mass of an object refers to the amount of matter it contains, measured in kilograms (kg) in the International System of Units (SI). - In the context of de Broglie's equation, a larger mass will result in a smaller wavelength, assuming the velocity remains constant. This inverse relationship shows why larger objects have wavelengths that are typically undetectable. 2. **Velocity \( v \):** - Velocity is the speed an object travels in a specific direction, usually measured in meters per second (m/s) in SI units. - As velocity increases, if the mass is held constant, the de Broglie wavelength decreases. This again reinforces the idea of the inverse relationship between wavelength and the product of mass and velocity. Therefore, in calculations involving de Broglie's equation for a car or other large objects, the small de Broglie wavelength often calculated (like the one in the solution) confirms that these wave properties are generally negligible at macroscopic scales.

Unit Conversion (Velocity)

Unit conversion is a fundamental skill that is essential when working with formulas in physics, such as the de Broglie wavelength equation. Converting units properly ensures that all variables align correctly for calculations. When given a velocity in kilometers per hour (km/hr), it's vital to convert this into meters per second (m/s) for consistency with the SI units used in the de Broglie equation. Here's a breakdown of converting from km/hr to m/s:- **Convert kilometers to meters:** Since 1 km equals 1000 meters, multiply the velocity in km/hr by 1000 to change it to meters/hr.- **Convert hours to seconds:** With 1 hour equaling 3600 seconds, divide by 3600 to convert meters per hour to meters per second. For instance, converting 36 km/hr:\[ 36 \text{ km/hr} = \frac{36 \times 1000}{3600} \text{ m/s} = 10 \text{ m/s} \] This step is crucial because it aligns the velocity with the units of mass and Planck's constant in the de Broglie equation. Accurate conversions ensure reliable and meaningful results when using physical equations in real-world scenarios.