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) \((\mathrm{dP} / \mathrm{d} \mathrm{V})_{\mathrm{T}}=\mathrm{K} / \mathrm{V}\) (c) \((\mathrm{dV} / \mathrm{dP})_{\mathrm{T}}=-\mathrm{K} / \mathrm{P}^{2}\) (d) \((\mathrm{dV} / \mathrm{dP})_{\mathrm{T}}=-\mathrm{K} / \mathrm{P}\)

Step-by-Step Solution

Answer

The correct answer is (c) \((dV/dP)_T = -K/P^2\).

Step 1: Understand Boyle's Law

Boyle's law states that for a given amount of gas at constant temperature, the pressure multiplied by the volume is constant. Mathematically, it can be expressed as \(PV = K\), where \(K\) is a constant when temperature is constant.

Step 2: Differentiate Boyle's Law with Respect to Volume

Differentiating \(PV = K\) with respect to volume \(V\), while holding temperature \(T\) constant, gives \(P + V \frac{dP}{dV} = 0\). Rearranging this gives \(\frac{dP}{dV} = -\frac{P}{V}\).

Step 3: Express \(\frac{dP}{dV}\) in Terms of \(K\)

Since \(PV = K\), we can substitute \(P\) using \(K/V\). This substitution gives \(\frac{dP}{dV} = -\frac{K}{V^2}\). We observe that the only expression matching the negative reciprocal dependence \(K/V\) in option \(b\) is incomplete given \(V\), hence more closely consider another term.

Step 4: Differentiate Boyle's Law with Respect to Pressure

Alternatively, differentiate \(PV = K\) with respect to pressure \(P\), while holding temperature \(T\) constant. This results in \(V + P \frac{dV}{dP} = 0\). Rearranging gives \(\frac{dV}{dP} = -\frac{V}{P}\).

Step 5: Express \(\frac{dV}{dP}\) in Terms of \(K\)

Using \(V = K/P\), we substitute for \(V\) and find \(\frac{dV}{dP} = -\frac{K}{P^2}\), which matches option \(c\).

Step 6: Final Step: Check Consistency with Options

Based on the calculations, the correct option that matches our differentiated equations using Boyle's Law is \(c\), \((dV/dP)_T = -K/P^2\). None of the other options align exactly with such solutions calculated from the law's principles.

Key Concepts

Differentiation of Gas Laws

Differentiation of gas laws involves using calculus to explore relationships between different gas law variables. In the context of Boyle's Law, this involves transforming the formula to understand how changes in one variable affect another when all other factors remain constant.

Boyle's Law is a foundational concept in chemistry and states that for a constant temperature, the pressure of a gas is inversely proportional to its volume. This means as the volume decreases, the pressure increases, and vice versa. To explore this with calculus, we differentiate the equation of Boyle's Law, \(PV = K\), in relation to volume \(V\), keeping temperature \(T\) constant.

Differentiating \(PV = K\) with respect to \(V\) gives us the equation \(P + V \frac{dP}{dV} = 0\). Rearranging this formula results in \(\frac{dP}{dV} = -\frac{P}{V}\). This differentiation reveals how a small change in volume leads to a change in pressure, providing insight into the behavior of gases under different conditions.

• Basic understanding helps in predicting gas behavior in various scientific and industrial processes.
• These calculations form the foundation for more complex models involving the interaction of gases.

Constant Temperature in Gas Laws

In the study of gas laws, the term "constant temperature" often refers to a condition where the temperature of a gas remains the same even when other factors like pressure or volume change. This scenario is widely used in conjunction with Boyle's Law.

When gas is contained in a rigid container where no heat is added or removed, the temperature remains constant. In this condition, any alteration in pressure will directly have an inverse effect on the volume. This constancy allows us to simplify the equations and predict the behavior of gases.

For example, if you have a balloon filled with air and you squeeze it without letting any air out, you increase the pressure inside the balloon while decreasing its volume. However, if the temperature is constant, the relationship between pressure and volume remains consistent, allowing you to predict one if you know the other.

• Constant temperature helps simplify calculations of gas behavior.
• It reveals insights about the energy states of gas molecules.
• This condition is also crucial when constructing pressure-volume graphs, aiding in visualizing gas behaviors.

Pressure-Volume Relationship

The pressure-volume relationship is central to understanding how gases behave under varying conditions. According to Boyle's Law, the pressure of a gas is inversely related to its volume when the temperature remains constant. This means that if the pressure increases, the volume decreases proportionally, and vice versa.

This inverse relationship can be expressed mathematically by the formula \(PV = K\), where \(K\) is a constant value for a given amount of gas at a set temperature. This formula tells us that the product of pressure and volume at constant temperature always equals the same number, \(K\).

Graphically, if you plot pressure against volume, you'd obtain a curve called a hyperbola. Such a graph is valuable in visualizing the behaviors of real gases as well as providing practical application in areas like engineering and meteorology.

• Understanding this relationship is crucial in fields such as chemistry and physics.
• It provides foundational knowledge necessary for advanced study in thermodynamics.
• Real-world applications include predicting how a gas behaves in enclosed spaces.