Problem 51

Question

The following statement(s) is (are) correct : [1999 - 3 Marks] (a) A plot of $\log K_{p}$ versus $1 / T$ is linear (b) A plot of $\log [X]$ versus time is linear for a first order reaction, $X \rightarrow P$ (c) A plot of $P$ versus $1 / T$ is linear at constant volume (d) A plot of $P$ versus $1 / V$ is linear at constant temperature

Step-by-Step Solution

Verified

Answer

Statements (a), (b), and (d) are correct.

1Step 1: Understanding the Statement (a)

The statement refers to the plot of $ \log K_{p} $ versus $ 1/T $. According to the van 't Hoff equation, \[ \log K_p = -\frac{\Delta H^\circ}{2.303R} \cdot \frac{1}{T} + \frac{\Delta S^\circ}{2.303R} \] where $ \Delta H^\circ $ and $ \Delta S^\circ $ are the standard enthalpy and entropy changes, respectively. This equation forms the equation of a straight line, $ y = mx + c $, implying the plot is linear.

2Step 2: Understanding the Statement (b)

For a first-order reaction $ X \rightarrow P $, the concentration $[X]$ decreases exponentially with time. The integrated rate law for a first-order reaction is \[ \log [X] = -kt + \log [X]_0 \] where $[X]_0$ is the initial concentration and $k$ is the rate constant. This is of the form $ y = mx + c $, indicating a linear plot of $ \log [X] $ versus time.

3Step 3: Understanding the Statement (c)

The statement involves a plot of $ P $ versus $ 1/T $ at constant volume. For an ideal gas, $ P = nRT/V $. Assuming volume is constant, $ P \propto T $. Thus, $ P $ versus $ 1/T $ would not be a straight line; instead, it will be a hyperbolic curve. Thus, this statement is not correct.

4Step 4: Understanding the Statement (d)

For an ideal gas at constant temperature, the relation $ PV = nRT $ indicates that $ P \propto 1/V $ (holding $ n $, $ R $, and $ T $ constant). Therefore, the plot of $ P $ versus $ 1/V $ is indeed linear, obeying the equation $ y = mx + c $.

Key Concepts

van 't Hoff EquationFirst-Order ReactionsIdeal Gas Law

van 't Hoff Equation

The van 't Hoff equation is a fundamental relation in chemical thermodynamics connecting the change in temperature with the equilibrium constant of a reaction. It is given by the formula: \[ \log K_p = -\frac{\Delta H^\circ}{2.303R} \cdot \frac{1}{T} + \frac{\Delta S^\circ}{2.303R} \] where:

$K_p$ is the equilibrium constant in terms of partial pressures.
$\Delta H^\circ$ is the standard change in enthalpy.
$\Delta S^\circ$ is the standard change in entropy.
$R$ is the universal gas constant.
$T$ is the temperature in Kelvin.

This equation implies that if you plot $\log K_p$ against $1/T$, the result is a straight line. The slope of this line is directly related to the enthalpy change of the reaction, and the intercept is related to the entropy change. This linearity is essential for determining these thermodynamic parameters experimentally.
Understanding this relation helps predict how shifts in temperature will affect the equilibrium position of a reaction, providing critical insight into reaction dynamics.

First-Order Reactions

In chemical kinetics, first-order reactions are reactions where the rate is proportional to the concentration of a single reactant. The rate expression for a first-order reaction can be written as: \[ \frac{d[X]}{dt} = -k[X] \] where:

$[X]$ is the concentration of the reactant.
$k$ is the rate constant of the reaction.

Upon integration, the equation becomes: \[ \log [X] = -kt + \log [X]_0 \] where $[X]_0$ is the initial concentration of the reactant.
This equation indicates that if you plot $\log [X]$ as a function of time, you will obtain a straight line. The slope of this line is $-k$, which provides a means to determine the rate constant experimentally.
Understanding first-order reaction kinetics is essential for predicting how the concentration of a reactant will change over time, which is vital in fields like pharmacokinetics and various chemical processes.

Ideal Gas Law

The Ideal Gas Law is a crucial equation in chemistry that links pressure, volume, and temperature of an ideal gas through the equation: \[ PV = nRT \] where:

$P$ is the pressure of the gas.
$V$ is the volume of the gas.
$n$ is the number of moles of the gas.
$R$ is the ideal gas constant.
$T$ is the temperature in Kelvin.

This law helps in determining the properties of gases in various chemical processes and is used to understand how gases will behave when subject to changes in conditions.
For the statement analyzed, we can look at the relationship $P \propto 1/V$ when temperature and moles of gas are constant, which leads us to another key relationship:

A plot of $P$ versus $1/V$ is linear when temperature is constant, in line with the ideal gas law.
Conversely, a plot of $P$ versus $1/T$ at constant volume is hyperbolic, which doesn't conform to a linear relationship.

Grasping the ideal gas law allows for informed predictions about gas behavior in chemical reactions and laboratory conditions.

Problem 50

Problem 54

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