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) P/RT (d) \(\mathrm{RT} / \mathrm{P}\)

Step-by-Step Solution

Answer

The correct answer is option (c) \( \frac{P}{RT} \).

Step 1: Understanding the Ideal Gas Law

The Ideal Gas Law is expressed as \( PV = nRT \), 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 expressed in Kelvin. We need to determine the number of moles per litre.

Step 2: Rewriting the Equation for Moles

To find the number of moles per litre, we want to solve the Ideal Gas Law for \( n/V \). Divide both sides of the equation by \( V \) and \( T \) to isolate \( n/V \): \( \frac{n}{V} = \frac{P}{RT} \).

Step 3: Identify the Correct Option

Now that we have \( \frac{n}{V} = \frac{P}{RT} \), compare this with given options: (a) \( \frac{PT}{R} \), (b) \( PRT \), (c) \( \frac{P}{RT} \), (d) \( \frac{RT}{P} \). The correct form is option (c) \( \frac{P}{RT} \).

Key Concepts

Gas Constant

The gas constant, often represented by the symbol \( R \), is a crucial component of the Ideal Gas Law. It's a universal constant that helps relate the energy scale to the temperature scale. Knowing about the gas constant allows us to relate other physical properties of the gas more effectively.

• The value of \( R \) in \((J \, mol^{-1} \, K^{-1})\) is approximately \( 8.314 \, J \, mol^{-1} \, K^{-1} \).
• It also can be expressed in different units, depending on the system used. For instance, \( R \) can be \( 0.0821 \, L \, atm \, mol^{-1} \, K^{-1} \) when dealing with pressure in atmospheres and volume in liters.

The gas constant bridges the proportional relationships between pressure, volume, and temperature in gas laws. Its versatile units allow scientists to work comfortably across various contexts, ensuring that calculations range seamlessly whether in chemical reactions or atmospheric studies.

Moles per Litre

Understanding how to calculate moles per litre is important in chemistry because it defines the concentration of a solution or a gas in a given volume. In the context of an ideal gas, this concept is a bit special. Knowing how to get from moles to moles per litre (We take the Ideal Gas Law \( PV = nRT \) and adjust it to determine moles per litre. By isolating \( n/V \), we can transform the formula to \( \frac{n}{V} = \frac{P}{RT} \), which allows us to express the number of moles per unit volume. This step ensures that we're handling precise amounts of substances when engaged in reactions or processes.

• Helps in determining the molar concentration in a reaction.
• Essential for understanding yield and reaction rates in chemical equations.

By expressing moles in terms of pressure, volume, and temperature, we understand how conditions affect the substance's behavior during reactions.

Pressure and Temperature Relationship

The relationship between pressure and temperature forms a fundamental pillar in the study of gases, and is encapsulated in the gas laws. According to the Ideal Gas Law, pressure and temperature are directly related as long as the volume and number of moles remain constant.When the pressure of a gas increases, its temperature also increases if the volume is held constant. Conversely, if the temperature decreases, so does the pressure. This relationship is intrinsic in scenarios where a gas is held in a rigid container.

• Expressed as \( P \propto T \), where \( \propto \) denotes proportionality, assuming constant volume & moles.
• Makes use of absolute temperature scales such as Kelvin due to their linear nature.

This proportionality tells us how gases behave under different conditions and is crucial in predicting the behavior of gases during thermal processes. Furthermore, understanding this relationship aids in the safe design of systems like pressurized gas reactors where temperature changes can influence pressure significantly.