Problem 61 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following is the correct relation between enthalpy of vaporization (Joule/mole) and boiling point of a liquid? (a) \(\Delta \mathrm{H}=1.5 \mathrm{P}_{\mathrm{b}}\) (b) \(\Delta \mathrm{H}=77 \mathrm{~T}\) (c) \(\Delta \mathrm{H}=88 \mathrm{~T}_{\mathrm{b}}\) (d) \(\Delta \mathrm{H}=0.5 \mathrm{~T}_{\mathrm{b}}\)

Step-by-Step Solution

Answer

Option (c) \( \Delta H = 88 T_b \) is the correct relation.

Step 1: Understand the Problem

We want to find the correct relation between the enthalpy of vaporization, \( \Delta H \), and the boiling point of a liquid, often denoted as \( T_b \). The boiling point is typically given in Kelvin, and the enthalpy of vaporization is expressed in Joules per mole. We need to match these terms in the given options.

Step 2: Know the Standard Equation

Recall that generally, the relation connecting enthalpy of vaporization and temperature is known by the Clausius-Clapeyron equation, which implies a form similar to \( \Delta H = a \cdot T_b \) for some proportional constant \( a \). We will use this relationship to validate the given options.

Step 3: Analyze Each Option

Evaluate each given option: - **Option (a)**: \( \Delta H = 1.5 P_b \) involves pressure, which is not the primary variable linked directly with \( \Delta H \) here.- **Option (b)**: \( \Delta H = 77 T \) suggests a temperature dependence, but it uses \( T \) instead of \( T_b \).- **Option (c)**: \( \Delta H = 88 T_b \) aligns with our expectation, where \( \Delta H \) is directly proportional to the boiling point \( T_b \).- **Option (d)**: \( \Delta H = 0.5 T_b \) also maintains a proper form but seems to lack a typical constant magnitude.

Step 4: Compare the Relationships

Options (c) and (d) present a form consistent with the expected dependence on \( T_b \). However, option (c) is more reasonable given the typical magnitude of constant as seen in similar thermodynamic equations involving temperature and energy relationships.

Step 5: Make a Conclusion

Based on the above analysis, option (c) has the correct form and likely correct magnitude as it expresses \( \Delta H \) in relation to the boiling point \( T_b \) in a plausible thermodynamic context. Thus, \( \Delta H = 88 T_b \) is considered the correct option.

Key Concepts

Clausius-Clapeyron equation

The Clausius-Clapeyron equation is a vital principle in understanding the phase transition between liquid and gas. It's most frequently applied in boiling phenomena, where it describes how the vapor pressure of a liquid relates to its temperature. This equation is particularly significant in thermodynamics because it gives insight into how pressure changes with temperature for a substance undergoing a phase change.
The Clausius-Clapeyron equation can be expressed as follows:\[\frac{dP}{dT} = \frac{\Delta H}{T \cdot \Delta V}\]where:

• \(dP\) is the change in pressure
• \(dT\) is the change in temperature
• \(\Delta H\) is the enthalpy of vaporization
• \(\Delta V\) is the change in volume of the substance

This equation essentially bridges concepts of pressure, temperature, and heat capacity by detailing how systems approach equilibrium during a phase change. Understanding this relationship clarifies why certain substances have specific boiling points, linking them directly to energy changes.

boiling point

The boiling point of a liquid is a critical thermodynamic property. It identifies the temperature at which a liquid turns into vapor. For a given substance, this point is constant under specific pressure conditions, typically at 1 atmosphere (101.3 kPa). At this temperature, the vapor pressure of the liquid equals the external pressure, allowing bubbles to form within the liquid.
Key concepts related to boiling point include:

• Vapor Pressure: When a liquid is heated, its molecules gain kinetic energy, increasing vapor pressure. At the boiling point, this pressure equals the atmospheric pressure.
• Normal Boiling Point: This is measured under standard atmospheric pressure. It's inherent to the substance's physical properties.
• Role in Phase Change: A boiling point marks the transition between liquid and gas phases. Energy absorbed at this temperature, known as the enthalpy of vaporization, is used to break intermolecular forces.

The boiling point is an essential concept for understanding the conditions under which substances vaporize, linking it with the enthalpy of vaporization as seen in the given exercise's options.

temperature dependence

Temperature dependence describes how a substance's properties change with varying temperature. In the context of boiling point and enthalpy of vaporization, temperature significantly impacts the behaviors and characteristics of substances.
For example, the enthalpy of vaporization, \(\Delta H\), is inherently dependent on a substance's temperature. Substances have unique kinetic energy distributions at different temperatures, affecting vapor pressure and phase changes:

• Higher Temperatures: Typically increase molecular activity, leading to higher vapor pressure and faster transition to the gas phase.
• Lower Temperatures: Reduce molecular movement, requiring more energy input to achieve vaporization.

In the exercise, option (c) highlights temperature dependence where enthalpy of vaporization is proportional to the boiling point. This suggests a direct relationship between energy required for phase change and temperature. Understanding this dependency helps predict how substances behave under various thermal conditions—an essential aspect in both scientific research and practical applications.