Problem 50 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following statement is correct? (a) A plot of \(\log k_{0}\) vs \(1 / \mathrm{t}\) is linear (b) A plot of log [X] vs time is linear for a first-order reaction, \(\mathrm{X} \longrightarrow \mathrm{P}\) (c) A plot of log \(\mathrm{P}\) vs \(1 / \mathrm{t}\) is linear at constant volume (d) A plot of P vs \(1 / \mathrm{V}\) is linear at constant pressure

Step-by-Step Solution

Answer

(d) A plot of P vs \(1 / V\) is linear at constant pressure.

Step 1: Understand the Meaning of Each Statement

Let's go through each statement one by one to identify what they imply and whether they are correct.(a) A plot of \( \log k_{0} \) vs \( 1 / \mathrm{t} \) is linear. - This statement suggests a relationship between the logarithm of an initial rate constant \( k_{0} \) and the reciprocal of temperature \( \mathrm{t} \). Typically, if a reaction follows the Arrhenius equation \( k = Ae^{-E_a/RT} \), plotting \( \log k \) versus \( 1/T \) gives a linear graph. However, this statement is about \( k_{0} \) and \( 1/\mathrm{t} \), where \( \mathrm{t} \) is undefined here, which makes this statement questionable.(b) A plot of log [X] vs time is linear for a first-order reaction, \( \mathrm{X} \longrightarrow \mathrm{P} \).- In first-order reactions, \( \ln [X] \) vs time is linear, not \( \log [X] \) vs time, because the integrated rate law for a first-order reaction is \( [X] = [X]_0 e^{-kt} \), and \( \ln [X] = \ln [X]_0 - kt \).(c) A plot of \( \log \mathrm{P} \) vs \( 1 / \mathrm{t} \) is linear at constant volume.- This statement seems incorrect. The conditions for linearity with pressure would relate more typically to inverse temperature in terms of reaction rates or gas laws, not directly as stated here.(d) A plot of P vs \( 1 / \mathrm{V} \) is linear at constant pressure.- This statement relates to Boyle's Law, \( PV = nRT \), which says at constant temperature and moles of gas, a plot of \( P \) versus \( 1/V \) is linear.

Step 2: Verify Against Known Principles

Let's cross-examine each statement against well-known principles of chemistry:- Statement (a): The correct form should involve temperature as \( 1/T \) when discussing the Arrhenius plot with \( k \), not \( 1/\mathrm{t} \), where \( \mathrm{t} \) is not clear.- Statement (b): An error is present as it confuses \( \log \) with \( \ln \). The linear form in a plot corresponds to \( \ln [X] = \ln [X]_0 - kt \), thus, when using natural logarithm, it aligns with first-order reactions.- Statement (c): This seems misleading without further context; pressure inversely depends on temperature, but not directly correlating with \( \log \mathrm{P} \) versus \( 1/\mathrm{t} \) across typically defined systems.- Statement (d): This aligns correctly with Boyle's Law, indicating it’s likely the correct statement given classical gas law relationships.

Step 3: State the Correct Answer

Based on common chemical principles like Boyle's Law, we gather:- Statement (d) indeed underlines Boyle's Law accurately. A plot of pressure, \( P \), versus the inverse of volume, \( 1/V \), is linear at constant temperature, upholding classical definitions.

Key Concepts

First-Order Reaction

In chemistry, a first-order reaction is a type of chemical reaction where the rate depends linearly on only one reactant concentration. This means that the rate of reaction is directly proportional to the concentration of one of the reactants. For a reaction of the form \( X \rightarrow P \), where \( X \) converts into product \( P \), the rate law can be expressed as \( -\frac{d[X]}{dt} = k[X] \), where \( k \) is the rate constant and \([X]\) is the concentration of reactant \( X \).

The integrated rate equation for a first-order reaction is \( [X] = [X]_0 e^{-kt} \), where \([X]_0\) is the initial concentration of \( X \), \( k \) is the rate constant, and \( t \) is time. To determine if a reaction is first-order, you can plot the natural logarithm of the concentration of the reactant, \( \ln[X] \), against time. This should yield a straight line with a negative slope, \( -k \).

Understanding first-order reactions is crucial because they often describe processes such as radioactive decay, many cellular reactions, and other significant chemical processes. Notice that while plotting \( \log [X] \) (using the logarithm to base 10) versus time might not yield a linear graph, using the natural logarithm \( \ln [X] \) will. This distinction is important for accurately determining reaction kinetics.

Arrhenius Equation

The Arrhenius equation is a key formula in chemical kinetics that describes how the rate constant \( k \) of a reaction depends on temperature. Developed by Svante Arrhenius, this equation provides insight into the effects of temperature on reaction rates and allows chemists to estimate the change in rate constant with temperature. The formula is given by:
\[ k = A e^{-E_a / RT} \]
where:

• \( k \) is the rate constant of the reaction.
• \( A \) is the pre-exponential factor, which is a constant for each chemical reaction.
• \( E_a \) is the activation energy required to initiate the reaction.
• \( R \) is the universal gas constant \( (8.314 \text{ J/mol K}) \).
• \( T \) is the absolute temperature in Kelvin.

Plotting \( \ln k \) versus \( 1/T \) yields a straight line with a slope of \( -E_a/R \) and an intercept of \( \ln A \), allowing determination of both activation energy and the pre-exponential factor. This linear relation between \( \ln k \) and \( 1/T \) provides a useful method for assessing the temperature dependence of reaction rates.

The Arrhenius equation not only helps predict reactions' behavior with temperature changes but also aids in understanding the concept of activation energy. It implies that an increase in temperature typically increases the reaction rate, as molecules possess more energy to overcome the activation energy barrier. This concept is central in fields such as chemical engineering, biochemistry, and environmental science.

Boyle's Law

Boyle's Law is a fundamental principle in chemistry that describes the relationship between the pressure and volume of a gas at constant temperature. Formulated by Robert Boyle in the 17th century, it states that the pressure of a gas is inversely proportional to its volume, assuming the temperature and the number of gas particles remain constant. Mathematically, it is represented as:
\[ PV = constant \]
where:

• \( P \) is the pressure of the gas.
• \( V \) is the volume of the gas.

This means that if you have a container with a gas and you reduce its volume by compressing it, the pressure will increase, provided the temperature doesn't change. Similarly, increasing the volume decreases the pressure.

This relationship can be visualized by plotting pressure against the inverse of volume (\( P \) versus \( 1/V \)), resulting in a linear graph. This graphical representation helps to solidify understanding and confirming the behavior described by the law.

Boyle's Law is crucial in understanding how gases behave under different physical conditions and is applied in various fields, including chemistry, physics, and engineering. Whether in the design of engines, the working of a syringe, or understanding breathing mechanisms, Boyle's Law is foundational in explaining and predicting the behavior of gases in practical applications.