Problem 12 - A Complete Resource Book in Chemistry for JEE Main

Question

Internal energy (E) and pressure of a gas of unit volume are related as (a) \(\mathrm{P}=\frac{2}{3} \mathrm{E}\) (b) \(\mathrm{P}=\frac{3}{2} \mathrm{E}\) (c) \(\mathrm{P}=\frac{\mathrm{E}}{2}\) (d) \(\mathrm{P}=2 \mathrm{E}\)

Step-by-Step Solution

Answer

(a) \(\mathrm{P}=\frac{2}{3} \mathrm{E}\)."

Step 1: Understand the Relationship

The internal energy \(E\) and pressure \(P\) of a gas of unit volume are related by the ideal gas law for certain conditions. In kinetic theory, for a monoatomic ideal gas in three dimensions, the internal energy \(E\) per unit volume is associated with pressure. Recall that for an ideal gas, \(P + \frac{2E}{3} = 0\).

Step 2: Use Kinetic Theory

According to kinetic theory, the internal energy \(E\) of a monoatomic ideal gas is given by \(E = \frac{3}{2}nRT\), where \(n\) is moles per unit volume and \(RT\) is the energy per mole. The pressure \(P\) is provided by \(P = nRT\).

Step 3: Relate Internal Energy and Pressure

From kinetic theory, \(E = \frac{3}{2} nRT\) and \(P = nRT\). Substitute the expression of \(nRT\) from \(P\) into \(E\), we get: \(E = \frac{3}{2}P\). Rearranging gives \(P = \frac{2}{3}E\).

Step 4: Select the Correct Option

Based on the relationship derived from kinetic theory, \(P = \frac{2}{3}E\) corresponds to option (a). Therefore, the correct answer is option (a).

Key Concepts

Ideal Gas Law

The Ideal Gas Law is a fundamental equation relating various properties of an ideal gas, providing insights into how gases behave under different conditions. This law combines several established relationships in physics and chemistry, such as Boyle's law, Charles's law, and Avogadro's law. The Ideal Gas Law is expressed as:\[PV = nRT\]

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

This equation shows that the product of pressure and volume is proportional to the temperature and amount of gas. It is a powerful tool because it allows scientists and engineers to predict how a change in one property (like pressure) will affect others (like temperature and volume). However, it's important to remember that it idealizes real gas behavior and is most accurate under low pressure and high temperature conditions.

Internal Energy

Internal energy refers to the total energy contained within a system as a result of its particle movements and interactions. In thermodynamics, understanding internal energy is crucial because it influences the state and behavior of the system, particularly in gases.For a monoatomic ideal gas, internal energy \(E\) is derived from its kinetic energy, which is directly related to the temperature of the gas. The expression for internal energy is given as:\[E = \frac{3}{2}nRT\]This equation indicates that the internal energy depends on:

• The number of moles \(n\) (which tells us how much gas is present),
• The Ideal Gas Constant \(R\), and
• The temperature \(T\) in Kelvin.

It's essential to grasp that the internal energy increases with temperature, implying that when a gas is heated, its particles move more rapidly, increasing their kinetic energy and thus, the total internal energy of the gas.

Monoatomic Ideal Gas

A monoatomic ideal gas is a simplified model of gas molecules that assumes particles composed of single atoms without complex interactions. In such gases, the atoms act as hard spheres moving randomly, and their behavior can be explained adequately by kinetic theory.Kinetic theory is crucial in understanding a monoatomic ideal gas. It provides that the energy of the gas solely comes from the translational motion of its particles, unlike diatomic or polyatomic gases, which also include rotational and vibrational motions. Thus, for monoatomic gases, the expression for internal energy and kinetic theory calculations become much less complex.One key relationship derived from the kinetic theory for monoatomic ideal gases is:\[E = \frac{3}{2}PV\]This tells us that the internal energy \(E\) is directly proportional to the pressure \(P\) and volume \(V\) of the gas. Such relationships allow physicists and chemists to predict how these gases will react under various conditions, making this model essential for academics and industry alike.