Problem 202 - A Complete Resource Book in Chemistry for JEE Main

Question

Rate constant of the first-order reaction when initial concentration \(\mathrm{C}_{\mathrm{o}}\) and concentration \(\mathrm{C}_{\mathrm{t}}\) at time \(\mathrm{t}\) is given by equation \(k_{\mathrm{t}}=\log \mathrm{C}_{0}-\log \mathrm{C}_{\mathrm{t}}\) Graph is a straight line if we plot (a) \(\mathrm{t} \operatorname{vs} \log \mathrm{C}_{0}\) (b) \(\mathrm{t}\) vs \(\log \mathrm{C}_{\mathrm{t}}\) (c) \(\mathrm{t}^{-1}\) vs \(\log \mathrm{C}_{\mathrm{t}}\) (d) \(\log \mathrm{C}_{\mathrm{o}}\) vs \(\log \mathrm{C}_{\mathrm{t}}\)

Step-by-Step Solution

Answer

The graph is a straight line if we plot (b) \( t \) vs \( \log C_t \).

Step 1: Understand the first-order reaction equation

Recall that for a first-order reaction, the rate constant \( k \) can be written as \( k = \frac{1}{t} \ln \frac{C_0}{C_t} \). We also know that \( \ln \frac{C_0}{C_t} = \ln C_0 - \ln C_t \). In chemistry, we often use the common log (\(\log_{10}\)) relationship \( \ln x = 2.303 \log_{10} x \). This means the rate constant can also be represented as \( k = \frac{2.303}{t} (\log C_0 - \log C_t) \). The question simplifies this to \( k_t = \log C_0 - \log C_t \) without the factor \( \frac{2.303}{t} \), which indicates a direct logarithmic relationship can be plotted.

Step 2: Identify the format for a linear graph

For the graph of a line to be linear, it should fit the equation of a straight line, \( y = mx + c \), where \( y \) and \( x \) are variables of the function, \( m \) is the slope, and \( c \) is the y-intercept. In this modified equation context, \( \log C_t = \log C_0 - kt \), where \( \log C_t \) could be treated as the dependent variable, \( y \), \( t \) as the independent variable, \( x \), with \( -k \) as the slope, and \( \log C_0 \) as the y-intercept.

Step 3: Match with the given options

Since \( \log C_t \) is set as \( y \) and \( t \) as \( x \) in the linear relation derived (\( \log C_t = \log C_0 - kt \)), the graph of \( t \) versus \( \log C_t \) follows a straight line with \( t \) as the independent variable and \( \log C_t \) as the dependent variable.

Key Concepts

Rate Constant

In a first-order reaction, the rate constant, often denoted as \( k \), is a fundamental parameter that helps describe how quickly a reaction occurs. Understanding this concept is crucial for analyzing reaction rates.
First-order reactions depend directly on the concentration of one reactant. This means that the rate of the reaction is proportional to the concentration of that reactant. The equation for a first-order reaction is:

• \( k = \frac{1}{t} \ln \frac{C_0}{C_t} \)

Here, \( C_0 \) represents the initial concentration and \( C_t \) is the concentration at any time \( t \). Where \( \ln \) indicates the natural logarithm, a mathematical way of expressing exponential relationships.
This equation can often be converted to:

• \( k = \frac{2.303}{t} (\log C_0 - \log C_t) \)

using the relationship between natural logarithm and common logarithm. Understanding this formula allows chemists and students to calculate how fast a reaction is proceeding and to compare the rates of different reactions under various conditions.

Logarithmic Plot

Analyzing reaction kinetics often involves creating plots that help visualize data for easier interpretation. A common plotting method for first-order reactions is the logarithmic plot.
When we consider a plot of \( t \) versus \( \log C_t \), it typically yields a straight line. This

• Linear relationship shows how the concentration of a reactant changes over time.

In this plot,

• \( t \) is the independent variable (x-axis),
• \( \log C_t \) is the dependent variable (y-axis).

This results in a linear equation format, \( y = mx + c \), where:

• \( -k \) is the slope (m)
• \( \log C_0 \) is the y-intercept (c).

This graphical method provides a simple way to determine the rate constant \( k \) of a first-order reaction by measuring the slope of the straight line, allowing for better insights and predictions about the reaction's behavior over time.

Concentration-Time Relationship

The concentration-time relationship in first-order reactions is important because it helps to predict how long a reaction might take and the concentration of remaining reactants over time.
In first-order reactions, the concentration of a substance decreases exponentially with time. This relationship is expressed as:

• \( \log C_t = \log C_0 - kt \)

By setting \( \log C_t \) as our y-variable and time, \( t \), as our x-variable, we can visualize this decreasing concentration linearly, further simplifying analysis.
This kind of analysis serves many practical applications where reaction efficiency and timing are critical.

• Pharmaceuticals: predicting how drugs degrade over time.
• Environmental science: understanding pollutant decay in lakes and rivers.

Having a grasp of this relationship enables us to optimize processes, control reaction conditions, and improve safety protocols across various fields, reinforcing its significance in both academic and practical applications.