Problem 164 - A Complete Resource Book in Chemistry for JEE Main

Question

Asuming that water vapour is an ideal gas, the internal energy change \((\Delta U)\) when 1 mol of water is vapourized at 1 bar pressure and \(100^{\circ} \mathrm{C}\), (Given: Molar enthalpy of vaporization of water at 1 bar and \(373 \mathrm{~K}\) \(=41 \mathrm{~kJ} \mathrm{~mol}^{-1}\) and \(\mathrm{R}=8.3 \mathrm{~J} \mathrm{~mol}^{-1} \mathrm{~K}^{-1}\) ) will be (a) \(3.7904 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (b) \(37.904 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (c) \(41.00 \mathrm{~kJ} \mathrm{~mol}^{-1}\) (d) \(4.100 \mathrm{~kJ} \mathrm{~mol}^{-1}\)

Step-by-Step Solution

Answer

The internal energy change when 1 mole of water is vaporized is 37.904 kJ/mol.

Step 1: Understanding the relationship between enthalpy and internal energy

The relationship between change in enthalpy (hhfluencefssssThlSynergspasssIo) in a process is given by the equation:\[\Delta H = \Delta U + P \Delta V\]Where:- \(\Delta H\) is the change in enthalpy,- \(\Delta U\) is the change in internal energy,- \(P\) is the pressure, and- \(\Delta V\) is the change in volume.

Step 2: Using Ideal Gas Equation for Volume Change

Assuming water vapor behaves as an ideal gas, the change in volume when 1 mole of liquid water is vaporized can be understood using the ideal gas equation. At the conversion point, \[PV = nRT,\]where \(n = 1 \text{ mol}\), \(R = 8.3 \text{ J mol}^{-1} \text{ K}^{-1}\), and \(T = 373 \text{ K}\). Using the given pressure \(P = 1 \text{ bar} = 10^5 \text{ Pa}\),\[\Delta V = \frac{RT}{P} = \frac{(8.3)(373)}{10^5} \text{ m}^3\].

Key Concepts

Internal Energy Change

In thermodynamics, the internal energy change (\( \Delta U \)) of a system is a pivotal concept for understanding various energy transformations. It represents the total change in energy within a system as it undergoes a process, which could involve heat transfer, work done by or on the system, or both.

When water vaporizes from liquid to gas, the internal energy change is tied closely to both the process conditions and the properties of water. The ideal gas law helps to determine the change in volume (\( \Delta V \)) as water turns into vapor under constant pressure and temperature conditions, allowing us to calculate the internal energy change using the equation:

• \( \Delta H = \Delta U + P \Delta V \)

By knowing the molar enthalpy of vaporization (\( \Delta H \)) and using data about pressure and volume from the ideal gas equation, we can derive \( \Delta U \), offering insight into this critical transformation.
Remember, \( \Delta U \) is often less intuitive than temperature changes or heat inputs, so recognize it accounts for energy stored within molecular bonds and interactions.

Enthalpy of Vaporization

The enthalpy of vaporization is a key concept in understanding phase changes, specifically when a substance transitions from a liquid to a gaseous state.Enthalpy, represented by \( H \), is essentially the total heat content of a system.

The molar enthalpy of vaporization given in this problem, \( 41 \text{ kJ mol}^{-1} \), indicates the amount of energy required to convert 1 mole of water from liquid at its boiling point to vapor.
A key equation connecting enthalpy change with internal energy change is:

• \( \Delta H = \Delta U + P \Delta V \)

The significance of vaporization lies in its ability to absorb significant amounts of energy, resulting in a phase change without a temperature change. Understanding this concept not only aids in comprehending energy changes during vaporization, but also plays a role in broader applications like heating systems and meteorology.

Thermodynamics in Chemistry

Thermodynamics forms the backbone of chemical reactions, helping us understand how energy changes accompany chemical processes. In chemistry, thermodynamic principles explain how energy is absorbed or released, providing insight into reaction feasibility and behavior.

In this context, the vaporization of water serves as a classic thermodynamic example where enthalpy and internal energy changes must be thoroughly understood.
The principles governing these changes are rooted in:

• First law of thermodynamics: Energy cannot be created or destroyed, only transformed.
• Equations of state, like the ideal gas equation, enabling predictions of change under specified conditions.

Additionally, chemical thermodynamics can predict reaction directions, estimate equilibrium positions, and calculate energy changes under varying pressures and temperatures. By mastering these fundamentals, students can better understand not only simple processes like vaporization but also complex chemical reactions imperative for fields like chemical engineering, environmental science, and energy production.