Problem 104 - A Complete Resource Book in Chemistry for JEE Main

Question

The following statement (s) is (are) correct (1) A plot of log KP versus \(1 / \mathrm{T}\) is linear (2) A plot of log \((\mathrm{X})\) versus time is linear for a first order reaction \(\mathrm{X} \longrightarrow \mathrm{P}\) (3) A plot of log P versus \(1 / \mathrm{T}\) is linear at constant volume (4) A plot of P versus \(1 / V\) is linear at constant temperature. (a) 1,2 (b) 2,4 (c) 2,3 (d) 1,4

Step-by-Step Solution

Answer

Options (a) 1,2 and (d) 1,4 are correct.

Step 1: Evaluate Statement 1

For statement 1, consider the Van't Hoff equation, which describes how the equilibrium constant \( K_P \) varies with temperature \( T \): \[ \log K_P = -\frac{\Delta H^\circ}{R} \cdot \frac{1}{T} + \frac{\Delta S^\circ}{R} \] This is a linear equation of the form \( y = mx + b \), where the slope \( m \) is \(-\frac{\Delta H^\circ}{R}\). Therefore, a plot of \( \log K_P \) versus \( \frac{1}{T} \) is indeed linear.

Step 2: Evaluate Statement 2

For a first-order reaction \( X \rightarrow P \), the rate equation is \[ \ln [X] = -kt + \ln [X_0] \] where \([X_0]\) is the initial concentration and \(k\) is the rate constant. This is a linear equation of the form \( y = mx + b \), indicating that a plot of \( \ln [X] \) versus time \( t \) is linear. Therefore, statement 2 is correct.

Step 3: Evaluate Statement 3

Statement 3 suggests that a plot of \( \log P \) versus \( 1/T \) is linear at constant volume. From the ideal gas equation \( PV = nRT \), at constant volume and number of moles the pressure \( P \) is proportional to temperature \( T \): \( P = \frac{nR}{V}T \). Therefore, \( \log P \) versus \( 1/T \) does not form a linear relationship, making this statement incorrect.

Step 4: Evaluate Statement 4

According to Boyle's Law, at constant temperature, the pressure of a gas is inversely proportional to its volume: \( P \propto \frac{1}{V} \). Therefore, a plot of \( P \) versus \( 1/V \) would be linear. This makes statement 4 correct.

Step 5: Identify Correct Options

Now that we have concluded, statements 1, 2, and 4 are correct. Referring to the given options, option (a) includes statements 1 and 2, while option (d) includes statements 1 and 4. Since both option (a) and option (d) include only correct statements, these are the correct choices.

Key Concepts

Van't Hoff equation

The Van't Hoff equation is essential for understanding how equilibrium constants change with temperature. This equation is given by: \[\log K_P = -\frac{\Delta H^\circ}{R} \cdot \frac{1}{T} + \frac{\Delta S^\circ}{R}\]which is a linear form akin to \(y = mx + b\).

• \(K_P\) represents the equilibrium constant based on partial pressures.
• \(\Delta H^\circ\) is the standard enthalpy change.
• \(R\) is the universal gas constant.
• \(T\) is the absolute temperature.
• \(\Delta S^\circ\) is the standard entropy change.

This equation illustrates that plotting \(\log K_P\) against \(\frac{1}{T}\) results in a straight line. The slope of that line is given by \(-\frac{\Delta H^\circ}{R}\), which indicates the enthalpy change's role in this relationship. Understanding the Van't Hoff equation helps in predicting how chemical equilibrium shifts with temperature changes.

first-order reaction

In the realm of chemical kinetics, a first-order reaction is one where the rate depends linearly on the concentration of one reactant. For a simple first-order reaction like \(X \rightarrow P\), the equation describing this process can be written as:\[\ln [X] = -kt + \ln [X_0]\]Here,

• \([X]\) is the concentration of reactant \(X\).
• \(k\) is the rate constant.
• \([X_0]\) is the initial concentration of \(X\).

This equation implies a straight line when you plot \(\ln [X]\) versus time \(t\), where the slope is equal to \(-k\). Once you understand the setup, predicting how the concentration changes over time becomes straightforward. This knowledge is foundational in determining reaction rates and timescales for chemical processes.

Boyle's Law

Boyle’s Law is a fundamental principle in chemistry that describes how the pressure (\(P\)) of a gas is inversely related to its volume (\(V\)) at a constant temperature. Mathematically, it is expressed as:\[P \propto \frac{1}{V}\]This means that as the volume of a gas decreases, its pressure increases proportionally, provided the temperature remains constant.When considering Boyle’s Law in visualization, plotting pressure (\(P\)) against the inverse of volume (\(\frac{1}{V}\)) gives a linear relationship. This graph comforts itself as evidence of the inverse relationship between pressure and volume, making it easier to predict behaviors of gas under different volume changes.Understanding Boyle’s Law is crucial in various real-world applications, such as breathing, pneumatic systems, and understanding how gases behave in different environmental conditions.

equilibrium constant

The equilibrium constant (\(K\)) is a pivotal concept in understanding chemical equilibria. This value helps determine the extent to which a reaction will proceed and what the concentrations of products and reactants will be at equilibrium. An equilibrium constant can be based on different concentration measures, such as pressure (\(K_P\)) or concentrations (\(K_c\)).

• A large equilibrium constant indicates a reaction with a significant formation of products.
• A small equilibrium constant suggests that reactants are favored at equilibrium.

This concept is invaluable in predicting the position of equilibrium under certain conditions. For plotting and prediction purposes, such as utilizing the Van't Hoff equation, the equilibrium constant allows for understanding how temperature impacts the reaction's direction and extent. Overall, grasping equilibrium constants is central to mastering chemical equilibria and reaction dynamics.