Problem 54 - A Complete Resource Book in Chemistry for JEE Main

Question

An ideal gas expands according to \(\mathrm{PV}=\) constant. On expansion, the temperature of gas (a) will rise (b) will drop (c) will remain constant (d) cannot be determined because the external pressure is not known

Step-by-Step Solution

Answer

The temperature will remain constant.

Step 1: Understanding Ideal Gas Expansion

The problem states that an ideal gas expands, and its pressure-volume product is constant (PV = constant). This hints at a particular kind of thermodynamic process.

Step 2: Recognizing the Process

The equation PV = constant represents an isothermal process for an ideal gas. In an isothermal process, the temperature remains constant as the gas expands.

Step 3: Applying Isothermal Process Knowledge

In an isothermal expansion of an ideal gas, since the temperature remains constant by definition, we can confirm what happens to the temperature as the gas expands.

Step 4: Conclusion

Given that this is an isothermal process, the temperature of the gas during expansion will remain unchanged.

Key Concepts

Isothermal Process

In the world of thermodynamics, an **isothermal process** is when a system experiences a change yet its temperature remains unchanged throughout the occurrence. For an ideal gas, this typically means that as the gas expands or is compressed, the internal energy remains constant.
This fascinating behavior occurs because any heat involved in the process will either be absorbed or released to the surroundings, ensuring the temperature remains stable. For example, when you slowly expand a gas in a balloon, if this takes place in an environment of constant temperature, it is undergoing an isothermal process.
This isothermal condition can be particularly useful in understanding energy changes without being distracted by temperature fluctuations, simplifying many calculations in thermodynamic problems.

Pressure-Volume Relationship

The **pressure-volume relationship** is a key component in understanding thermodynamics, especially within isothermal processes. This relationship is mathematically defined by Boyle's Law, which states that for a given amount of gas at constant temperature, the pressure and volume are inversely proportional.
Mathematically, this is represented by the formula: \[ PV = \text{constant} \] where \( P \) is the pressure of the gas, \( V \) is its volume, and the constant depends on the amount of gas and its temperature.
Essentially, this means if you increase the volume of the gas (let's say, by pulling a piston), the pressure inside the gas must decrease if the temperature stays constant. Conversely, reducing the volume should cause an increase in pressure.
Understanding this inverse relationship helps us predict the behavior of gases under various conditions and is fundamental for solving many problems related to the ideal gas law.

Thermodynamic Processes

Thermodynamic processes describe how systems change in terms of their energetics and equilibria when subject to different variables like temperature, pressure, and volume. These processes can be categorized based on how they handle these variables.

• **Isothermal process:** As previously explained, temperature remains constant while pressure and volume change.
• **Adiabatic process:** These processes occur without transferring heat to or from the system, leading to changes in internal energy reflected by temperature changes.
• **Isobaric process:** Pressure stays constant, allowing for changes in volume and temperature.
• **Isochoric process:** Volume remains constant, and any heat transfer converts directly to a change in internal energy, affecting pressure and temperature.

Each type of thermodynamic process introduces unique characteristics and relationships between variables, helping us understand the limitations and capabilities of energy conversions in natural and engineered systems. By mastering these concepts, you gain valuable insights into how energy systems work, which is crucial for advanced studies in physics and engineering.