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 P, temperature \(\mathrm{T}\) and gas constant \(\mathrm{R}\) is [2002] (a) \(\mathrm{PT} / \mathrm{R}\) (b) PRT (c) \(\mathrm{P} / \mathrm{RT}\) (d) RT/P

Step-by-Step Solution

Answer

(c) \( \frac{P}{RT} \) is the number of moles per litre for an ideal gas.

Step 1: Understand the Ideal Gas Law

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: Solve for Moles per Litre

Since we need the number of moles per litre, start by rearranging the Ideal Gas Law equation to solve for \( n/V \), which represents moles per litre \( \left( \frac{n}{V} \right) \). Divide both sides of the equation by \( RT \) to get \( n/V = P/RT \).

Step 3: Select the Correct Option

From step 2, we have \( \frac{n}{V} = \frac{P}{RT} \). Comparing this with the given options, the correct answer is option (c) \( \frac{P}{RT} \).

Key Concepts

Moles per Litre

Moles per litre is an important concept in understanding the concentration of gases in a given volume. When dealing with gases, we often want to know how many moles of gas are present in a certain volume, such as a litre. This concept is crucial for calculations involving chemical reactions in gaseous systems.
To calculate moles per litre using the Ideal Gas Law, we start with the equation:

• \( PV = nRT \)

Here, \( n/V \) represents the number of moles per litre. By rearranging the equation and solving for \( n/V \), we find:

• \( \frac{n}{V} = \frac{P}{RT} \)

This equation tells us that the moles per litre of an ideal gas can be determined by dividing the pressure \( P \) by the product of the gas constant \( R \) and the temperature \( T \). Understanding this relationship allows us to predict how changes in conditions will affect concentration in a gaseous system.

Gas Constant

The gas constant \( R \) is a fundamental parameter in the Ideal Gas Law and other gas-related equations. It serves as a bridge between the units of pressure, volume, temperature, and the amount of substance in a gas.
The value of the gas constant is typically given as \( R = 0.0821 \) L atm mol⁻¹ K⁻¹ when pressure is in atmospheres. This constant ensures that the equations involving gases remain consistent across different units of measurement.
Key points about \( R \):

• It correlates the physical properties of a gas.
• Helps in predicting how gases behave under different conditions.
• Is essential in converting between macroscopic measurements (like pressure and volume) and microscopic quantities (like moles).

By using \( R \) in the Ideal Gas Law, one can calculate unknown properties of gases when other conditions are known.

Pressure and Temperature Relationship

Pressure and temperature are intimately related in the context of gases, as described by the Ideal Gas Law. According to this law, pressure \( P \) and temperature \( T \) are directly proportional when the volume and amount of gas are kept constant.
As temperature increases, the energy of gas molecules increases, which often causes an increase in pressure. Conversely, as temperature decreases, so does pressure if everything else remains constant.
To understand this relationship better, consider:

• \( PV = nRT \) implies \( P \propto T \) if \( n \) and \( V \) are constant.
• Increasing temperature results in quicker molecule movement, leading to higher pressure.
• A decrease in temperature results in slower molecular motion, hence lower pressure.

Understanding this relationship can help predict and control conditions in processes involving gases, such as chemical reactions and industrial applications.