Problem 66 - A Complete Resource Book in Chemistry for JEE Main

Question

The enthalpy of vaporization of a liquid is \(30 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and entropy of vaporization is \(5 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}\). The boiling point of the liquid at 1 atm is (a) \(250 \mathrm{~K}\) (b) \(400 \mathrm{~K}\) (c) \(450 \mathrm{~K}\) (d) \(600 \mathrm{~K}\)

Step-by-Step Solution

Answer

The boiling point is 600 K (option d).

Step 1: Understanding the Gibbs Free Energy Change

Begin by recalling the equation for the Gibbs free energy change for phase transition: \( \Delta G = \Delta H - T \Delta S \). At the boiling point, the liquid and vapor phases are in equilibrium, which implies \( \Delta G = 0 \). Thus, \( \Delta H = T \Delta S \) at equilibrium.

Step 2: Substitute Given Values

Use the given values for the enthalpy (\( \Delta H = 30 \mathrm{~kJ~mol}^{-1} = 30000 \mathrm{~J~mol}^{-1} \)) and the entropy (\( \Delta S = 5 \mathrm{~J~mol}^{-1} \mathrm{~K}^{-1} \)) of vaporization in the equilibrium equation: \( 30000 = T \times 5 \).

Step 3: Solve for Temperature

Rearrange the equation from Step 2 to solve for the temperature \( T \): \[ T = \frac{30000}{5} \].

Step 4: Calculation of Boiling Point

Calculate the temperature by performing the division: \[ T = 600 \mathrm{~K} \]. This temperature corresponds to the boiling point of the liquid at 1 atm.

Key Concepts

Enthalpy of Vaporization

The enthalpy of vaporization is a key concept when discussing the phase change from liquid to gas. It represents the amount of energy required to convert one mole of a liquid into a gas. This process involves breaking the intermolecular forces that hold the liquid together. For instance, in the original exercise, the enthalpy of vaporization is given as \( 30 \text{ kJ mol}^{-1} \). This means that 30 kilojoules of energy are needed for each mole of liquid to transition into its gaseous state.

Understanding enthalpy of vaporization is crucial because it provides insights into the strength of intermolecular forces within a liquid. Stronger forces mean higher energy — or enthalpy — is needed. This concept is fundamental in calculating boiling points and understanding the energy changes during vaporization.

Entropy of Vaporization

Entropy is a measure of the disorder or randomness in a system, and entropy of vaporization specifically looks at the change in disorder when a liquid becomes a gas. During vaporization, entropy increases since gas particles have more freedom and thus more disorder than those in a liquid. In the exercise, the entropy of vaporization is \( 5 \text{ J mol}^{-1} \text{ K}^{-1} \).

This value indicates how much the randomness of the system is altered by vaporizing one mole of liquid. Higher entropy of vaporization implies a larger increase in disorder. This concept is significant in thermodynamics as it helps in understanding how energy is spread out in a system. Furthermore, it plays a role in determining equilibrium conditions, such as boiling points, when combined with enthalpy values.

Gibbs Free Energy

Gibbs Free Energy, denoted as \( \Delta G \), is a thermodynamic function that combines enthalpy and entropy to determine whether a process will occur spontaneously. For phase transitions, like boiling, the equation \( \Delta G = \Delta H - T \Delta S \) is used. At equilibrium, like at the boiling point of a liquid, \( \Delta G = 0 \).

This balance tells us that the energy from the enthalpy change is precisely countered by the entropy contribution at a specific temperature. In the provided exercise, solving \( 0 = 30000 - T \times 5 \) allows us to find the boiling point. Understanding Gibbs Free Energy is essential as it gives insight into the feasibility and spontaneity of reactions and phase transitions.

Phase Equilibrium

Phase equilibrium occurs when the processes of phase change are in balance, meaning the rates of evaporation and condensation are equal. At this point, the system is at its boiling point, and no net change occurs in the amount of liquid or vapor.

In the context of our exercise, phase equilibrium is realized when the Gibbs Free Energy is zero, indicating an equilibrium between liquid and vapor phases. The temperature at which this occurs is the boiling point — calculated to be \( 600 \text{ K} \) from our earlier steps.

This concept is critical in understanding how systems stabilize and how various factors like pressure and temperature affect phase changes. Knowing the conditions for phase equilibrium helps predict and control the behavior of liquids and gases in various applications.