Problem 7 - Arihant AIEEE Physics

Question

The pressure of a gas kept in an isothermal container is \(200 \mathrm{kPa}\). If half the gas is removed from it, the pressure will be: (a) \(100 \mathrm{kPa}\) (b) \(200 \mathrm{kPa}\) (c) \(400 \mathrm{kPa}\) (d) \(800 \mathrm{kPa}\)

Step-by-Step Solution

Answer

The new pressure is 100 kPa (option a).

Step 1: Understand the Problem

We have an isothermal container, which means the temperature of the gas does not change. The initial pressure of the gas is given as \(200 \mathrm{kPa}\). We need to find the new pressure when half of the gas is removed from the container.

Step 2: Recall the Ideal Gas Law

The ideal gas law is given by \( PV = nRT \), where \( P \) is pressure, \( V \) is volume, \( n \) is the amount of substance (in moles), \( R \) is the universal gas constant, and \( T \) is the temperature. In an isothermal process, \( T \) remains constant.

Step 3: Apply the Relationship of Pressure and Moles

Under isothermal conditions, and assuming constant volume, the pressure of a gas is directly proportional to the number of moles, i.e., \( P \propto n \). Therefore, if the number of moles is halved, the pressure will also be halved.

Step 4: Calculate the New Pressure

The initial pressure is \(200 \mathrm{kPa}\) and since the number of moles of gas is reduced to half, the pressure reduces to half as well. Thus, the new pressure is \( \frac{200}{2} = 100 \mathrm{kPa} \).

Key Concepts

Isothermal Process

An isothermal process refers to a thermodynamic change occurring in a system where the temperature remains constant. In the context of gases, it means that even though the volume and pressure might change, the temperature does not. This concept is vital in understanding the behavior of gases under specific conditions and is a key part of the Ideal Gas Law.

During an isothermal process, if a gas is allowed to expand, its pressure will decrease, provided the temperature is held constant. Conversely, if the gas is compressed, the pressure will increase. The constancy of temperature is crucial because it dictates that the internal energy change (ΔU = 0) is zero, thus simplifying analysis using the ideal gas law: \( PV = nRT \), where \(T\) is constant.

Understanding isothermal processes helps in solving problems involving changes in the state of gases while maintaining a fixed temperature.

Pressure of Gases

Pressure is a measure of the force exerted by the gas particles as they collide with the walls of their container. This concept is fundamental in gas laws and is typically expressed in units of kPa (kilopascals) or atm (atmospheres).

According to Boyle's law, which applies to isothermal conditions, the pressure of a gas is inversely proportional to its volume when the temperature remains constant. So, if a gas is compressed into a smaller volume, the pressure increases and vice versa.

In isothermal processes, as observed in our exercise, the pressure also directly correlates with the number of moles for a given volume when temperature is maintained constant. This relationship, described in the Ideal Gas Law, helps predict how changes in the quantity of gas influence the pressure.

Moles in Gas Laws

Moles refer to the amount of substance in a given sample of gas and play a crucial role in gas law calculations. The mole is a base unit in chemistry that provides a method to express amounts of a chemical substance.

In the context of the Ideal Gas Law \(PV = nRT\), \(n\) represents the number of moles. Moles are key to understanding the proportionality of gases; the more moles of gas present in a container, the greater the pressure if the volume and temperature are kept constant.

For an isothermal process involving a decrease in the number of moles, as in the given exercise where half the gas is removed, the pressure is directly proportional to the number of moles, meaning the pressure also reduces to half. This understanding simplifies calculations and predictions in practical scenarios involving gas reactions and storage.