Problem 145 - A Complete Resource Book in Chemistry for JEE Main

Question

For an ideal gas, number of mole per litre in terms of its pressure \(\mathrm{P}\), temperature \(\mathrm{T}\) and gas constant \(\mathrm{R}\) is [2002] (a) \(\mathrm{PT} / \mathrm{R}\) (b) PRT (c) \(\mathrm{P} / \mathrm{RT}\) (d) \(\mathrm{RT} / \mathrm{P}\)

Step-by-Step Solution

Answer

Option (c) \( \frac{P}{RT} \) is correct.

Step 1: Understanding the Ideal Gas Equation

The ideal gas law is given by the equation: \( PV = nRT \), where \( P \) is the pressure, \( V \) is the volume, \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Rearrange the Ideal Gas Law

Our goal is to find the number of moles per litre. Start by rearranging the equation to solve for \( n \): \( n = \frac{PV}{RT} \).

Step 3: Express Moles per Volume

Since we are interested in moles per litre, express \( \frac{n}{V} \) by dividing both sides of the rearranged equation by \( V \): \( \frac{n}{V} = \frac{P}{RT} \).

Step 4: Verify Against Given Options

Compare the expression \( \frac{n}{V} = \frac{P}{RT} \) to the given options. Option (c) \( \frac{P}{RT} \) matches our expression for moles per litre.

Key Concepts

Understanding the Number of Moles per Litre

Moles per litre is an important concept when working with the ideal gas law. It helps determine the concentration of a gas in terms of moles within a specific volume. The ideal gas law is a fundamental equation that relates pressure, volume, and temperature to the amount of gas in moles using the formula: \[ PV = nRT \] where:

• \( P \) represents pressure
• \( V \) represents volume
• \( n \) represents the number of moles
• \( R \) is the gas constant
• \( T \) is the temperature in Kelvin

To find the number of moles per litre, we want to express our equation in terms of moles per volume. By rearranging the ideal gas law, we solve for \( n \): \[ n = \frac{PV}{RT} \] Then, further simplify this to find moles per litre (moles per unit of volume): \[ \frac{n}{V} = \frac{P}{RT} \] This expression \( \frac{P}{RT} \) tells us how many moles of gas exist in each litre of volume.

Exploring Pressure-Temperature Relation

Pressure and temperature form a close relationship within the ideal gas law. This association is vital because it describes how changes in temperature affect the pressure of a gas, assuming the volume and quantity of gas remain constant. In the equation \( PV = nRT \), altering the temperature \( T \) while keeping the same amount of gas in a fixed volume will lead to variations in pressure \( P \). If the temperature rises, the pressure will correspondingly increase, a concept known as Gay-Lussac's law.Some important considerations regarding this relationship include:

• As temperature increases, molecules move faster, leading to an increase in pressure.
• Conversely, decreasing the temperature slows down the molecules, reducing pressure.

Overall, maintaining balance between these two properties is crucial in applications such as pressurized gas containers and weather systems, where the pressure-temperature relation aids in predicting and controlling conditions.

Unveiling the Gas Constant R

The gas constant \( R \) is an essential component of the ideal gas law. It enables the equation \( PV = nRT \) to equate the units of pressure, volume, and temperature with the number of moles of a gas. The gas constant \( R \) has a fixed value when proper units are applied: \[ R = 8.314 \, \text{J/mol} \, \text{K} \] This value represents energy per mole per Kelvin during conditions where one mole of an ideal gas is heated by one degree Kelvin.Key aspects of the gas constant include:

• Its universality for nearly all calculations involving ideal gases.
• Its role in ensuring dimensions remain consistent in the ideal gas equation.

Remember, \( R \)'s value varies depending on the chosen units. However, the value \( 8.314 \, \text{J/mol} \, \text{K} \) is commonly used across scientific disciplines owing to its basis in the SI unit system. This invaluable constant aids in simplifying complex relationships between gas properties, making it a core element in understanding ideal gas behaviors.