Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

The entropy change in the fusion of \(1 \mathrm{~mol}\) of a solid melting at \(27^{\circ} \mathrm{C}\) (Latent heat of fusion, \(2930 \mathrm{~J} \mathrm{~mol}^{-1}\) ) is (a) \(9.77 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (b) \(10.73 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (c) \(2930 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\) (d) \(108.5 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The entropy change is (a) \(9.77 \mathrm{JK}^{-1} \mathrm{~mol}^{-1}\).

Step 1: Understand the Concept of Entropy Change

Entropy change (9) is calculated when a phase change occurs using the formula: \( \Delta S = \frac{q_{\text{rev}}}{T} \), where \( q_{\text{rev}} \) is the latent heat of fusion and \( T \) is the temperature in Kelvin.

Step 2: Convert Temperature to Kelvin

Given the melting point at \( 27^{\circ} \mathrm{C} \), we need to convert this to Kelvin: \( T = 27 + 273 = 300 \text{ K} \).

Step 3: Apply the Entropy Change Formula

Substitute the given latent heat of fusion and the Kelvin temperature into the formula: \( \Delta S = \frac{2930 \mathrm{~J/mol}}{300 \mathrm{~K}} \).

Step 4: Calculate the Entropy Change

Perform the division to find \( \Delta S \): \( \Delta S = \frac{2930}{300} = 9.77 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \).

Step 5: Identify the Correct Answer Choice

The calculated entropy change is \( 9.77 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \). Compare this with the given options and select the matching answer: (a) \( 9.77 \mathrm{JK}^{-1} \mathrm{~mol}^{-1} \).

Key Concepts

Latent Heat of Fusion

When a substance changes from one state of matter to another, we call this a phase change. In the case of melting, a solid becomes a liquid. During this process, the heat required to change the phase without changing the temperature is called the latent heat of fusion. The term 'latent' refers to the "hidden" energy used to break the bonds that hold the solid structure together without raising the temperature. For example, to melt 1 mole of a solid at its melting point, you'll need a specific amount of energy, which is the latent heat of fusion. In our exercise, this value is given as 2930 J/mol. Understanding this concept is important because it explains why the temperature remains constant during the entire melting process despite energy being added.

Phase Change

A phase change is an important concept in understanding thermodynamics. It occurs when a substance transitions from one state of matter—solid, liquid, or gas—to another. These changes require or release energy, which often comes in the form of heat.
Examples of phase changes include:

• Melting: Solid to liquid.
• Freezing: Liquid to solid.
• Vaporization: Liquid to gas.
• Condensation: Gas to liquid.

During a phase change, the temperature of the substance remains constant. This happens because all the heat energy is used to alter the state rather than change the temperature. In our case, melting at a specific temperature, such as 27°C, is a clear example of a solid changing phase to liquid with energy added without changing the temperature.

Temperature Conversion

Temperature conversion is necessary to ensure all calculations are consistent and correct in physics and chemistry. Many formulas in thermodynamics require temperature in Kelvin, not Celsius. The Kelvin scale is an absolute temperature scale that starts at absolute zero, the coldest possible temperature.
To convert temperatures from degrees Celsius to Kelvin, simply add 273.15 to the Celsius temperature.
For example,

• 27°C + 273.15 = 300.15 K

In practice, you might see it rounded to 300 K for simplicity. This conversion allows us to use the correct units in thermodynamic equations to achieve accurate results. This conversion is essential for calculating the entropy change during phase changes or any process involving heat.

Thermodynamics Formula

In thermodynamics, various formulas describe the relationships between heat, work, and temperature. One key formula is for entropy change, especially during a phase change like melting.
The formula is given as:\[\Delta S = \frac{q_{\text{rev}}}{T}\]Where:

• \(\Delta S\) is the change in entropy.
• \(q_{\text{rev}}\) is the reversible heat added or removed (here, the latent heat of fusion).
• \(T\) is the absolute temperature in Kelvin.

Using this formula, you can calculate how the disorder in a system changes during a phase transition.
For instance, in our exercise, you divide the given latent heat of fusion (2930 J/mol) by the temperature in Kelvin (300 K) to find the entropy change of the system. Mastery of these formulas enables a deeper understanding of processes and transformations within a thermodynamic system.