Problem 140 - A Complete Resource Book in Chemistry for JEE Main

Question

The freezing point of isobutane is \(-160^{\circ} \mathrm{C} \cdot \Delta \mathrm{H}_{\text {(solid } \rightarrow \text { liquidd })^{-}}\) is \(+4520 \mathrm{~J} \mathrm{~mol}^{-1}\). For this fusion process, entropy change in \(\mathrm{J} \mathrm{mol}^{-1}\) is \(10 \mathrm{y}\). The value of \(\mathrm{y}\) is

Step-by-Step Solution

Answer

The value of \(y\) is 4.

Step 1: Identify Given Values

We are provided with the freezing point of isobutane as \(-160^{\circ} \mathrm{C}\), which is equivalent to\(-160 + 273 = 113 \ \mathrm{K}\). The given enthalpy change \(\Delta \mathrm{H}_{\text {solid} \rightarrow \text {liquid} })\) is \(+4520 \, \mathrm{J} \, \mathrm{mol}^{-1}\). This is the energy required to convert the solid phase to liquid at the freezing point.

Step 2: Relate Entropy Change to Enthalpy Change

The relationship between the entropy change \(\Delta S\) and the enthalpy change \(\Delta H\) during a phase change at a constant temperature \(T\) is given by the formula:\[\Delta S = \frac{\Delta H}{T}\]where \(\Delta S\) is the entropy change per mole in \(\mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\), \(\Delta H = 4520 \, \mathrm{J} \, \mathrm{mol}^{-1}\), and \(T = 113 \, \mathrm{K}\).

Step 3: Calculate the Entropy Change

Substitute the known values into the entropy change formula:\[\Delta S = \frac{4520 \, \mathrm{J} \, \mathrm{mol}^{-1}}{113 \, \mathrm{K}} = 40 \mathrm{J} \, \mathrm{mol}^{-1} \, \mathrm{K}^{-1}\]Since \(\Delta S = 10y\), we equate \(10y = 40\).

Step 4: Solve for \(y\)

From the equation \(10y = 40\), solve for \(y\):\[y = \frac{40}{10} = 4\]

Key Concepts

Phase Change

In thermodynamics, a phase change refers to the transition of a substance from one state of matter to another. These states include solid, liquid, and gas. When isobutane freezes or melts, it undergoes one such phase change. During a phase change, the temperature of the substance remains constant, even though energy is being added or removed.
There are several types of phase changes:

• Solid to liquid (melting)
• Liquid to solid (freezing)
• Liquid to gas (vaporization)
• Gas to liquid (condensation)
• Solid to gas (sublimation)
• Gas to solid (deposition)

These changes occur at specific temperatures and pressures, which are known as the substance's melting point, boiling point, and other characteristic temperatures. Each material has its unique temperatures for phase changes. For isobutane, the freezing occurs at a very low temperature of \(-160^{\circ} \mathrm{C}\), equivalent to \(113 \mathrm{K}\).

Enthalpy Change

Enthalpy change, \(\Delta H\), is a key concept in thermodynamics that represents the heat exchanged at constant pressure during a phase change. It provides insight into the energy changes involved when a substance undergoes a transformation in state.
For example, when isobutane changes from a solid to a liquid, \(\Delta H\) is the energy absorbed from the surroundings by the substance as it transitions to the liquid phase. This specific energy cost is known as the 'enthalpy of fusion'. For isobutane transitioning from its solid phase to the liquid, the enthalpy change is \(+4520 \mathrm{~J} \mathrm{~mol}^{-1}\), indicating that this amount of energy is required per mole of isobutane.
During a phase change, the enthalpy change allows us to understand the energy required and the way it affects the structure of molecules in the substance. It is crucial in calculating other thermodynamic variables, such as entropy change and temperature dependencies of reactions.

Specific Heat Capacity

Specific heat capacity is a measure of the amount of heat required to change the temperature of a substance by one degree Celsius (or one Kelvin per unit mass). While specific heat capacity does not directly account for enthalpy changes related to phase changes, it plays a significant role in determining how substances behave as they are heated or cooled.
Different phases of a substance (solid, liquid, gas) will generally have different specific heat capacities. This is why you might need more energy to heat a liquid than a solid, for example. Although specific heat capacity is more applicable to changes within a phase rather than at the points of phase change, it is an integral concept for understanding how substances absorb energy without changing phase.
Knowing the specific heat capacity can also help predict how a substance will absorb heat over time and become important as the substance approaches a phase change temperature. This fundamental knowledge is essential for applying thermodynamic principles effectively in real-world situations.

Thermodynamics

Thermodynamics is the branch of science that deals with heat, work, and the associated energy transformations. The principles of thermodynamics allow us to predict the direction in which energy flows and how different forms of energy are converted into one another during chemical and physical processes.
There are four laws of thermodynamics, but the two most relevant to phase and enthalpy changes are the first and second laws:

• First Law (Law of Energy Conservation): Energy cannot be created or destroyed, only converted from one form to another. This law underlines the enthalpy observations during a phase change where energy is transferred as heat.
• Second Law: Entropy of an isolated system always increases over time. This law governs the direction of heat transfer and the increased entropy during melting processes, such as with isobutane.

Understanding thermodynamics helps students grasp how and why substances change phase and how energy is required or released during these transformations. It sets the foundation for applying these principles in fields such as engineering, chemistry, and environmental sciences.