Problem 95 - A Complete Resource Book in Chemistry for JEE Main

Question

One mole of a non-ideal gas undergoes a change of state \((2.0 \mathrm{~atm}, 3.0 \mathrm{~L}, 95 \mathrm{~K}) \longrightarrow(4.0 \mathrm{~atm}, 5.0 \mathrm{~L}\) \(245 \mathrm{~K}\) ) with a change in internal energy, \(\Delta \mathrm{U}=30.0 \mathrm{~L}\) atm. The change in enthalpy \((\Delta H)\) of the process in \(L\) atm is (a) \(40.0\) (b) \(42.3\) (c) \(44.0\) (d) not defined, because pressure is not constant

Step-by-Step Solution

Answer

The change in enthalpy is 44.0 L atm (Option c).

Step 1: Identify Formula for Change in Enthalpy

The change in enthalpy \( \Delta H \) for a process can generally be calculated using the equation: \[ \Delta H = \Delta U + \Delta (PV) \] where \( \Delta U \) is the change in internal energy and \( \Delta (PV) \) is the change in the product of pressure and volume.

Step 2: Calculate Change in (PV)

Calculate \( \Delta (PV) \) using the initial and final states of the gas. The initial state is \((2.0 \text{ atm}, 3.0 \text{ L})\) and the final state is \((4.0 \text{ atm}, 5.0 \text{ L})\). \[ \Delta (PV) = (P_2 V_2) - (P_1 V_1) = (4.0 \text{ atm} \times 5.0 \text{ L}) - (2.0 \text{ atm} \times 3.0 \text{ L}) \] \[ \Delta (PV) = 20.0 - 6.0 = 14.0 \text{ L atm} \]

Step 3: Calculate Change in Enthalpy

Using the calculated values for \( \Delta U \) and \( \Delta (PV) \), substitute them into the enthalpy equation.\[ \Delta H = \Delta U + \Delta (PV) = 30.0 \text{ L atm} + 14.0 \text{ L atm} \]\[ \Delta H = 44.0 \text{ L atm} \]

Step 4: Conclusion

The change in enthalpy for the process is \( 44.0 \text{ L atm} \), which corresponds to option (c).

Key Concepts

Non-Ideal Gas

In real-world scenarios, gases often deviate from the ideal behavior described by the ideal gas law. This deviation results in what we call non-ideal gases. The key difference between ideal and non-ideal gases is that in non-ideal gases, interactions between gas molecules are significant. In non-ideal gases, these interactions and the volume occupied by the gas molecules themselves cannot be ignored, especially at high pressures and low temperatures. Unlike ideal gases, non-ideal gases follow the van der Waals equation, a more complex equation that accounts for molecular size and intermolecular forces. Understanding the behavior of non-ideal gases is crucial for accurate predictions and calculations in thermodynamic processes. It is particularly important when determining changes in properties like enthalpy during state changes, as seen in the exercise.

Internal Energy

Internal energy (94U) is a fundamental concept in thermodynamics, representing the total energy contained within a system. It includes kinetic energy (motion of molecules) and potential energy (interactions between molecules). When we discuss changes in a system, such as the non-ideal gas in the exercise, internal energy helps us understand the energy exchange at a molecular level. The change in internal energy, 94U, can be calculated during a thermodynamic process and is often associated with heat exchanges and work done by or on the system. In the given exercise, the change in internal energy was noted as 30.0 L atm, a crucial value used to determine the change in enthalpy. This shows how internal energy ranks as a cornerstone of the First Law of Thermodynamics, connecting heat and work to the energy dynamics within a system.

Pressure Volume Work

Pressure-volume work is an essential part of understanding how energy is transferred in thermodynamic processes involving gases. When a gas changes state, its volume and pressure can vary, and these variations can do work on or by the gas. The work done by or on a gas is calculated by the product of pressure and the change in volume, expressed as: \[W = P 94V\]In the exercise, the changes in pressure and volume led to the need to calculate this pressure-volume component to find the overall change in a system's state. By understanding pressure-volume work, one can appreciate how this energy transfer mechanism impacts the overall energy balance, influencing the calculation of enthalpy changes.

Thermodynamics

Thermodynamics is the science of energy transformations, focusing on how energy is converted into work and heat. It provides the framework for analyzing processes involving energy changes, like the transformation of a non-ideal gas from one state to another. This field of study is grounded in the laws of thermodynamics:

• The First Law, or Law of Energy Conservation, states that energy cannot be created or destroyed, only transformed.
• The Second Law deals with entropy, emphasizing that energy transformations are irreversible and that some energy is always dispersed.

In the exercise, thermodynamics principles are applied to determine changes in enthalpy, internal energy, and the pressure-volume product. This analysis exemplifies the broad applications of thermodynamics in solving real-world energy transformation problems, enabling us to predict and optimize energy exchanges in numerous processes.