Problem 47 - A Complete Resource Book in Chemistry for JEE Main

Question

According to Boyle's law (a) \((\mathrm{dP} / \mathrm{dV})_{\mathrm{T}}=-\mathrm{K} / \mathrm{P}^{2}\) (b) \((\mathrm{dP} / \mathrm{dV})_{\mathrm{T}}=\mathrm{K} / \mathrm{V}\) (c) \((\mathrm{dV} / \mathrm{dP})_{\mathrm{T}}=-\mathrm{K} / \mathrm{P}^{2}\) (d) \((\mathrm{dV} / \mathrm{dP})_{T}=-\mathrm{K} / \mathrm{P}\)

Step-by-Step Solution

Answer

The correct answer is (d) \((\mathrm{dV} / \mathrm{dP})_{T}=-\mathrm{K} / \mathrm{P}\).

Step 1: Identify Boyle's Law Equation

Boyle's law states that for a given amount of gas at constant temperature, the pressure of the gas is inversely proportional to its volume. This can be expressed mathematically as: \[ PV = K \]where \( P \) is the pressure, \( V \) is the volume, and \( K \) is a constant.

Step 2: Differentiate Boyle's Law

To find the differential relationship, we differentiate \( PV = K \) with respect to \( V \) at constant temperature \( T \):- Use the product rule: \( \frac{d}{dV}(PV) = P \cdot \frac{dV}{dV} + V \cdot \frac{dP}{dV} = 0 \)- Simplify to: \( P + V \frac{dP}{dV} = 0 \), which leads to \( \frac{dP}{dV} = -\frac{P}{V} \).

Step 3: Apply the Correct Option

Given the options in the problem, we are looking for the correct differential expression that matches the derived relation. Rearranging \( \frac{dP}{dV} = -\frac{P}{V} \) gives: \( \frac{dV}{dP} = -\frac{V}{P} \).Given options, (d) fits this expression with K being the proportional constant.

Key Concepts

Pressure-Volume Relationship

The pressure-volume relationship is a fundamental concept in gas laws, encapsulated in Boyle's Law. This principle states that when the temperature of a given amount of gas is held constant, the pressure of the gas is inversely proportional to its volume. Mathematically, this is expressed as:

• PV = K

where P represents the pressure, V represents the volume, and K is a constant value.
This means that if the volume of the gas decreases, the pressure increases, and vice versa, as long as the temperature remains unchanged.

This inverse relationship can be visualized graphically as a hyperbola, where each point on the graph represents a unique state of the gas with specific pressure and volume values that satisfy the equation. Understanding this relationship is crucial not only in chemistry but also in various engineering applications where gas behavior under different conditions is essential.

Differential Calculus in Chemistry

Differential calculus is a vital mathematical tool used in chemistry to understand and describe how changes in one variable influence another. In the context of Boyle's Law, we use differentiation to explore the relationship between pressure and volume under constant temperature conditions.

By differentiating the equation

• PV = K

with respect to volume, we aim to determine how a small change in volume (7;dV7;) affects the pressure (7;dP7;). We apply the product rule of calculus to find the derivative. This involves:

• First deriving the expression for the product of pressure and volume.
• Then setting the total derivative equal to zero because the system remains at constant temperature.
• Solving for \(\frac{dP}{dV}\) yields \(-\frac{P}{V}\).

These derivations allow chemists to predict how small shifts in volume can alter the pressure, offering a deeper insight into gas dynamics beyond simple proportional relationships. Understanding these changes supports more advanced studies and practical applications in chemical reactions and processes.

Gas Laws

Gas laws are a set of rules that describe how gases behave under different conditions of temperature, volume, and pressure. They are foundational principles in chemistry, explaining how changes in one of these variables affect the others.

Boyle's Law, as discussed, is just one part of a larger framework of gas laws, which includes:

• Charles's Law: Relates volume and temperature, stating that volume increases with temperature at constant pressure.
• Gay-Lussac's Law: Relates pressure and temperature, where pressure increases with temperature when volume is fixed.
• Avogadro's Law: Indicates that volume is directly proportional to the amount of gas at constant temperature and pressure.

These laws can be combined to derive the Ideal Gas Law, represented as \(PV = nRT\), where \(n\) denotes the number of moles and \(R\) is the gas constant.

By understanding these laws, students can gain comprehensive insights into how gases interact with their environments and apply these concepts to solve real-world problems in physics, engineering, and environmental studies.