Problem 7 - Master Resource Book in JEE Main Physics

Question

Photons absorbed in matter are converted to heat. A source emitting \(n\) photon/sec of frequency \(v\) is used to converting of ice at \(0^{\circ} \mathrm{C}\) to water at \(0^{\circ} \mathrm{C}\). Then, the time, \(T\) taken for the conversion (a) decreases with increasing \(n\), with \(v\) fixed (b) decreases with \(n\) fixed, \(v\) increasing (c) remains constant with \(n\) and \(v\) changing such that \(n v=\) constant (d) increases when the product \(n \underline{v}\) increases

Step-by-Step Solution

Answer

The correct options are (a), (b), and (c). Option (d) is incorrect.

Step 1: Understanding Energy Conversion

The energy of each photon is equal to \( E = h u \), where \( h \) is Planck's constant and \( u \) is the frequency. If a source emits \( n \) photons per second, then the total energy per second emitted is \( n \times h u \). This energy is absorbed by ice to convert it into water.

Step 2: Energy Required for Ice-Water Conversion

The energy required to convert ice at \(0^{\circ} \mathrm{C}\) to water at \(0^{\circ} \mathrm{C}\) is given by the latent heat formula \( Q = mL \), where \( m \) is the mass of the ice and \( L \) is the latent heat of fusion. This is the energy needed to carry out the conversion.

Step 3: Calculating Time for Conversion

The time \( T \) required for conversion of ice to water is given by \( T = \frac{mL}{n h u} \). This formula is derived by setting the energy absorbed equal to the energy needed: \( n h u \times T = mL \). Thus, \( T \) is inversely proportional to both \( n \) and \( u \).

Step 4: Analyze Each Option

- Option (a): Increasing \( n \) (more photons per second) decreases \( T \) as \( n \) is in the denominator.- Option (b): Increasing \( u \) (higher frequency) decreases \( T \) as it also factors into the denominator.- Option (c): If \( n u \) is constant, \( T = \frac{mL}{n h u} \) will remain constant.- Option (d): Increasing the product \( n u \) decreases \( T \), not increases, as \( n u \) is the denominator factor.

Step 5: Conclusion

Options (a), (b), and (c) decrease or maintain \( T \) as described. However, option (d) is incorrect because increasing \( n u \) decreases time \( T \), not increases.

Key Concepts

Heat Conversion

When photons are absorbed by a substance, they often convert into other forms of energy, such as heat.
Heat conversion is a process where the absorbed energy from photons raises the thermal energy—or heat—of a material.
This increase in heat can change the state of a material, such as melting ice into water without increasing temperature.
The effectiveness of converting photon energy into heat is crucial in processes like changing the state of matter at a constant temperature.
In our exercise, photons are used to melt ice at 0°C, meaning their energy is turned into the heat needed to change ice into water.

Latent Heat of Fusion

The latent heat of fusion is the amount of energy required to change a solid (like ice) into a liquid (like water) without changing its temperature.
This energy is measured in Joules per kilogram (J/kg) and acts at the molecular level to overcome forces holding the structure of the solid.
It is important because, during a phase change, even when temperature remains constant, energy is still required to change the state.

• For water, the latent heat of fusion is approximately 334,000 J/kg.
• Without sufficient energy to achieve this transition, the phase change from solid to liquid cannot occur.

In our context, the photons emitted provide this energy, using their energy to complete the transformation from ice to water.

Photon Frequency

Photon frequency, denoted by \( v \), is a key concept in understanding the energy of a photon.
Frequency refers to the number of vibrations or waves per second, measured in Hertz (Hz).
The frequency of a photon determines its energy levels.

• Higher frequency photons have more energy, as expressed in the formula \( E = h v \), where \( h \) is Planck's constant.
• Thus, an increase in frequency leads to an increase in the energy each photon can transfer.

This is why increasing the frequency of photons can reduce the time required to convert ice to water, as more energy is provided with each photon.

Energy Conversion Formula

The energy conversion formula ties together the concepts of photon frequency and latent heat of fusion.
The energy per photon is represented as \( E = h v \), where \( h \) stands for Planck's constant and \( v \) is the frequency of the photon.
When discussing energy conversion, particularly in context with the latent heat, the relationship is reflected as \( Q = mL \).
Here, \( Q \) is the total energy required to convert ice into water at the given conditions, \( m \) is the mass, and \( L \) is the latent heat of fusion.

• In our exercise, we derive that \( T = \frac{mL}{n h v} \), illustrating that time \( T \) is inversely proportional to both photon emission rate \( n \) and frequency \( v \).
• Thus, increasing either \( n \) or \( v \) shortens the conversion time, as more energy is transferred in a given period.

This comprehensive view illuminates how efficiently photons can be used to manipulate and change the state of materials.