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 X\longrightarrow \(\longrightarrow\) (3) A plot of log P versus \(1 / \mathrm{T}\) is linear at constant volume (4) A plot of P versus \(1 / \mathrm{V}\) 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), (2), and (3), making the answer (c) 2, 3.

Step 1: Analyze Statement 1

The statement suggests a linear relationship between \( \log K_P \) and \( \frac{1}{T} \). Using the Van't Hoff equation, which is \[ \ln K_P = -\frac{\Delta H}{R} \cdot \frac{1}{T} + \text{constant} \], this indicates a linear plot with a negative slope. Thus, statement (1) is correct.

Step 2: Analyze Statement 2

Statement 2 suggests \( \log (X) \) versus time is linear for a first-order reaction. For first-order kinetics, the relationship is given by \( \log [A] = -kt + \log [A]_0 \), which describes a linear plot. Therefore, statement (2) is correct.

Step 3: Analyze Statement 3

Statement 3 postulates a linear relationship between \( \log P \) and \( \frac{1}{T} \) at constant volume. This follows from the Arrhenius equation for pressure assuming ideal gas behavior and consistent reaction enthalpy. Like statement (1), this leads to a linear plot, making statement (3) correct.

Step 4: Analyze Statement 4

Statement 4 proposes \( P \) versus \( \frac{1}{V} \) is linear at constant temperature. Using Boyle's law (\( PV = \text{constant} \)), when temperature is constant, rearranging gives \( P = \frac{\text{constant}}{V} \), which is not linear. Thus, statement (4) is incorrect.

Step 5: Identify the Correct Options

From the analysis, statement (1), (2), and (3) are correct. Therefore, the correct option would be (c) 2, 3, as option (d) includes statement (4), which we found to be incorrect.

Key Concepts

Van't Hoff Equation

The Van't Hoff equation is a fundamental relationship in chemical thermodynamics that allows us to understand how equilibrium constants change with temperature. The equation is written as:

• \( \ln K_P = -\frac{\Delta H}{R} \cdot \frac{1}{T} + \text{constant} \)

Here, \( K_P \) is the equilibrium constant in terms of pressure, \( \Delta H \) is the change in enthalpy, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

The equation indicates that plotting \( \ln K_P \) against \( 1/T \) will result in a straight line with a slope of \( -\Delta H/R \). If the slope is negative, it suggests that the reaction is exothermic, releasing heat as it proceeds.

This linear relationship helps chemists predict how a chemical reaction will respond to changes in temperature, making it easier to control and optimize reaction conditions in industrial and laboratory settings.

First-Order Reaction

First-order reactions are types of chemical reactions where the rate depends linearly on the concentration of one reactant. This means the reaction rate is directly proportional to the concentration of the reactant.

The mathematical expression describing a first-order reaction is:

• \( \log [A] = -kt + \log [A]_0 \)

Where \([A]\) is the concentration of the reactant at time \( t \), \([A]_0\) is the initial concentration, and \( k \) is the rate constant.

This equation results in a straight-line plot when \( \log [A] \) is plotted against time, showing that the rate of the reaction decreases linearly as the concentration diminishes. First-order kinetics are especially common in nuclear decay and some reactions involving gases and solutions.

Boyle's Law

Boyle's Law is a principle in chemistry describing the inverse relationship between the pressure and volume of a gas at constant temperature. The formal equation for Boyle's Law is:

• \( PV = \, \text{constant} \)

Here, \( P \) stands for pressure, \( V \) for volume, and the product is constant as long as temperature remains unchanged.

If you rearrange the equation, you will find that:

• \( P = \frac{\text{constant}}{V} \)

This means that pressure will decrease as volume increases and vice versa. However, it does not suggest a linear plot if \( P \) is graphed against \( 1/V \), highlighting the importance of the inverse relationship rather than a direct linear one.

Boyle's Law is a cornerstone in understanding gas behavior under compression or expansion conditions, with applications ranging from engineering to meteorology.

Arrhenius Equation

The Arrhenius equation is crucial for understanding how reaction rates vary with temperature. It provides insight into the effect of temperature on the speed of chemical reactions:

• \( k = Ae^{-E_a/(RT)} \)

Where \( k \) is the rate constant, \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the gas constant, and \( T \) is the temperature in Kelvin.

This equation can be rearranged into:

• \( \ln k = -\frac{E_a}{R} \cdot \frac{1}{T} + \ln A \)

When \( \ln k \) is plotted versus \( 1/T \), it results in a straight line whose slope is \( -E_a/R \). This linear relationship helps to determine the activation energy of the reaction.

The Arrhenius equation is a fundamental concept in chemical kinetics, widely used to predict how different conditions, especially temperature, can impact the rates of reactions—vital for designing everything from drug synthesis to materials engineering.