Answer in brief, any two questions. a. The latent heat of vaporization is much larger than the latent heat of fusion of a substance. Why? [2] b. Why does it take longer time to dry wet clothes in…

a. Latent heat of vaporization vs. latent heat of fusion

The latent heat of vaporization ($L_v$) is significantly larger than the latent heat of fusion ($L_f$) for a substance due to the following reasons:

1. Intermolecular Forces:

   • During fusion (melting), only the intermolecular forces holding the solid structure together must be overcome. These forces are relatively weaker.
   • During vaporization (boiling), all intermolecular forces must be broken, and the molecules must gain enough kinetic energy to escape into the gaseous phase. This requires much more energy.

2. Distance Between Molecules:

   • In fusion, the distance between molecules increases slightly (from solid to liquid).
   • In vaporization, the distance between molecules increases dramatically (from liquid to gas), requiring far more energy to overcome the increased separation.

3. Kinetic Energy Requirements:

   • For vaporization, molecules must achieve high kinetic energy to break free from the liquid’s surface tension and enter the vapor phase. This demands a much larger energy input compared to fusion.

Mathematical Insight: The energy required to change a substance from solid to liquid ($L_f$) is typically 10% of the energy required to vaporize it ($L_v$). For example, water has $L_f=334\text{kJ/kg}$ and $L_v=2260\text{kJ/kg}$.

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b. Drying clothes in rainy summer vs. sunny winter

Clothes dry faster in sunny winter than in rainy summer due to the following factors:

1. Evaporation Rate:

   • Sunny winter: The sun provides radiant heat, increasing the temperature of wet clothes. Higher temperature accelerates the kinetic energy of water molecules, causing faster evaporation.
   • Rainy summer: High humidity in the air reduces the evaporation rate because the air is already saturated with moisture. Even if the temperature is high, the water vapor cannot escape into the air easily.

2. Humidity Levels:

   • In summer, the relative humidity is high (often >70%), making it harder for water to evaporate from clothes.
   • In winter, the air is drier (lower humidity), allowing water to evaporate more efficiently.

3. Air Movement:

   • Sunny winter days often have gentle breezes that help carry away water vapor from the clothes.
   • Rainy summer days may have still or humid air, hindering evaporation.

4. Temperature Paradox:

   • While summer is hotter, the high humidity negates the effect of temperature on evaporation. Evaporation depends on the difference in vapor pressure between the liquid and the surrounding air, not just temperature.

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c. Internal energy of compressed vs. rarefied gas at the same temperature

The internal energy ($U$) of a compressed gas is less than that of a rarefied gas at the same temperature due to:

1. Intermolecular Potential Energy:

   • In a compressed gas, molecules are closer together, increasing the intermolecular attractive forces. This reduces the potential energy of the system because work is done on the gas to compress it.
   • In a rarefied gas, molecules are far apart, minimizing intermolecular forces. The potential energy is higher because no external work is done to push molecules together.

2. Work Done on the Gas:

   • Compressing a gas requires work to be done on it, which decreases its internal energy (since $\Delta U=Q-W$, and $W$ is positive for compression).
   • Rarefying a gas (expansion) absorbs work, increasing its internal energy.

3. Ideal Gas Assumption:

   • For an ideal gas, internal energy depends only on temperature ($U=\frac{3}{2}nRT$ for monatomic gases). However, real gases deviate from this due to intermolecular forces, especially at high pressures (compressed state).
   • At the same temperature, a compressed real gas has lower internal energy because some energy is stored as intermolecular potential energy (negative contribution).

Key Point: The internal energy of a real gas is given by: $$U=\frac{3}{2}nRT+U_{\text{potential}}$$ where $U_{\text{potential}}$ is negative for compressed gases (due to attractive forces) and less negative or zero for rarefied gases. Thus, $U_{\text{compressed}}<U_{\text{rarefied}}$ at the same temperature.