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 (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

The correct statements are 1 and 4, so the answer is (d) 1,4.

Step 1: Analyze Statement 1

For the first statement, the relationship between the equilibrium constant \( K_P \) and temperature \( T \) can be described by the Van't Hoff equation: \[ \ln K_P = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \] where \( \Delta H^\circ \) is the standard enthalpy change and \( \Delta S^\circ \) is the standard entropy change. This equation is indeed linear, so the statement is correct.

Step 2: Analyze Statement 2

For the second statement, consider a first-order reaction: the concentration of X decreases exponentially over time. The linear form is \( \ln [X] = \ln [X]_0 - kt \), where \( k \) is the rate constant. Hence, a plot of \( \ln [X] \) (not \( \log [X] \)) versus time is linear. Therefore, this statement is not correct because it uses \( \log (X) \), not \( \ln [X] \).

Step 3: Analyze Statement 3

The third statement concerns a plot of \( \log P \) versus \( 1/T \). When considering gas behavior under constant volume, Boyle's law or the ideal gas equation at constant volume doesn't directly relate pressure to \( 1/T \) linearly. Thus, this statement is not correct.

Step 4: Analyze Statement 4

In statement four, we consider the ideal gas law where \( PV = nRT \). At constant temperature \( T \), and fixed \( nR \), the equation rearranges to \( P = \frac{nRT}{V} \), showing that \( P \) is directly proportional to \( 1/V \). Therefore, a plot of \( P \) versus \( 1/V \) is indeed linear at constant temperature, making this statement correct.

Step 5: Determine Correct Choices

From the analysis above, statements 1 and 4 are correct. Therefore, the correct answer is option (d) 1,4.

Key Concepts

Van't Hoff Equation

The Van't Hoff equation provides a way to understand how the equilibrium constant \( K_P \) for a reaction changes with temperature. This is pivotal in the study of chemical thermodynamics, as it links the change in equilibrium position with the enthalpy and entropy changes of the system. The equation is expressed as: \[ \ln K_P = -\frac{\Delta H^\circ}{R} \left(\frac{1}{T}\right) + \frac{\Delta S^\circ}{R} \] Here's what each term means:

• \( \Delta H^\circ \) is the standard enthalpy change. It represents the overall change in heat during the reaction, measured in joules per mole.
• \( R \) is the universal gas constant, approximately equal to 8.314 J/(mol K).
• \( T \) is the temperature in Kelvin.
• \( \Delta S^\circ \) is the standard entropy change; it signifies the change in disorder or randomness.

The graph of \( \ln K_P \) versus \( 1/T \) results in a straight line. The slope of this line is \( -\Delta H^\circ /R \), and the intercept is \( \Delta S^\circ /R \). This linearity makes it easy to determine enthalpy and entropy changes from experimental data.

First Order Reactions

First order reactions are characterized by a rate that depends linearly on the concentration of one reactant. The general form for such a reaction is \( A \) converting to products, with a rate expression often simplified to: \[ \frac{d[A]}{dt} = -k[A] \] Where:

• \( [A] \) is the concentration of reactant \( A \)
• \( k \) is the rate constant, having units of \( s^{-1} \)

Upon integration, the rate equation transforms to its linear form: \[ \ln [A] = \ln [A]_0 - kt \] In this equation:

• \( [A]_0 \) is the initial concentration of \( A \)
• \( t \) is the time elapsed

A plot of \( \ln [A] \) versus time yields a straight line with a slope of \( -k \), indicating a clear linear relationship which distinguishes first order reactions from others. The original statement mentioned \( \log [X] \) instead of \( \ln [X] \), which is a key distinction since \( \log \) usually refers to base 10 while \( \ln \) indicates natural logarithm base \( e \). This accounts for the correction.

Ideal Gas Law

The Ideal Gas Law describes the behavior of ideal gases with the equation: \[ PV = nRT \] Each term in the equation has significant meaning:

• \( P \) is the pressure of the gas
• \( V \) is the volume it occupies
• \( n \) represents the number of moles of the gas
• \( R \) is the ideal gas constant, equal to 8.314 J/(mol K)
• \( T \) is the temperature in Kelvin

At constant temperature, this law demonstrates that there is an inverse relationship between pressure \( P \) and volume \( V \), which is known as Boyle's Law: \[ P \propto \frac{1}{V} \]. A plot of \( P \) versus \( 1/V \) under these conditions results in a linear graph, making this statement correct. This property is vital for understanding gas behaviors in different thermodynamic processes, particularly those at constant temperatures.