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 \((\Delta \mathrm{H})\) of the process in \(\mathrm{L}\) \(\mathrm{atm}\) 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 \( 44.0 \ \mathrm{L \ atm} \) (option c).

Step 1: Recall the Enthalpy Change Formula

The change in enthalpy \( \Delta H \) is expressed by 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 the Initial and Final PV Product

Calculate the initial \( PV \): \( 2.0 \ \mathrm{atm} \times 3.0 \ \mathrm{L} = 6.0 \ \mathrm{L \ atm} \). Calculate the final \( PV \): \( 4.0 \ \mathrm{atm} \times 5.0 \ \mathrm{L} = 20.0 \ \mathrm{L \ atm} \).

Step 3: Determine Change in PV

The change in \( PV \) is given by \( \Delta (PV) = 20.0 \ \mathrm{L \ atm} - 6.0 \ \mathrm{L \ atm} = 14.0 \ \mathrm{L \ atm} \).

Step 4: Calculate Change in Enthalpy

Now, substitute the known values into the enthalpy change formula: \( \Delta H = \Delta U + \Delta (PV) = 30.0 \ \mathrm{L \ atm} + 14.0 \ \mathrm{L \ atm} = 44.0 \ \mathrm{L \ atm} \).

Key Concepts

Enthalpy Change

Enthalpy change is a fundamental concept in thermodynamics that helps us understand how heat energy is absorbed or released during a chemical process. When a system experiences a change in state, such as a change in pressure, volume, or temperature, we can quantify the energy involved using enthalpy change, represented by the symbol \( \Delta H \).
The formula for calculating enthalpy change is given by:

• \( \Delta H = \Delta U + \Delta (PV) \)

Here, \( \Delta U \) represents the change in internal energy of the system, and \( \Delta (PV) \) denotes the change in the product of pressure and volume. This formula reflects that enthalpy is a state function, meaning it only depends on the initial and final states of the system, not the path taken to change states.
In practical terms, understanding enthalpy change is critical for processes like heating, cooling, and engine operations, where pressure and volume play significant roles in determining energy changes.

Non-Ideal Gas

Real-life gases, unlike ideal gases, often exhibit non-ideal behavior, meaning they do not always perfectly follow the Ideal Gas Law. A non-ideal gas is one where interactions between molecules and the volume occupied by the molecules themselves are significant enough to affect the gas's properties.
In the case of a non-ideal gas, factors such as high pressures or low temperatures can lead to deviations from the Ideal Gas Law. These deviations are often corrected using equations of state like the Van der Waals equation, which accounts for molecular interactions and finite volume.
To calculate thermodynamic properties, such as the change in internal energy or enthalpy for non-ideal gases, these equations are essential. We must take into account the real behaviors and properties these gases express, which are critical in chemical reactions, industrial processes, and natural phenomena.

Internal Energy

Internal energy is an intrinsic property of a system that includes all the kinetic and potential energy of its molecules. It is a crucial term in the first law of thermodynamics, which states that the energy of an isolated system is constant unless energy is added to or removed from the system.
The internal energy change, represented as \( \Delta U \), signifies how energy is stored or used within a system when it undergoes transitions from one state to another. Mathematically, internal energy change can often be calculated during reactions or processes as:

• \( \Delta U = Q - W \)

where \( Q \) is the heat exchanged, and \( W \) is the work done by or on the system.
Understanding internal energy helps us comprehend how a system's energy is affected by changes in temperature, volume, and pressure. This knowledge is pivotal in designing engines, refrigerators, and even in weather forecasting, where energy transformations are omnipresent.