Question

Which of the following statement(s) is/are incorrect? a. A plot of \(\mathrm{P}\) versus \(\mathrm{l} / \mathrm{V}\) is linear at constant temperature. b. A plot of P versus \(1 / \mathrm{T}\) is linear at constant volume. c. A plot of \(\log \mathrm{K}_{\mathrm{p}}\) versus \(1 / \mathrm{T}\) is linear d. A plot of \(\log [x]\) versus time is linear for zero order reaction.

Short answer

Statements (b) and (d) are incorrect.

Step-by-step solution

Analyze statement (a)

Statement (a) states that a plot of pressure \( P \) versus \( l/V \) is linear at constant temperature. According to Boyle's Law, \( P \propto 1/V \) at constant temperature, which implies a plot of \( P \) against \( 1/V \) is indeed linear. Therefore, this statement is correct.

Analyze statement (b)

Statement (b) suggests a plot of pressure \( P \) versus \( 1/T \) is linear at constant volume. This is not supported by the ideal gas law since when volume \( V \) is constant, pressure \( P \) is proportional to temperature \( T \) and should not linearly relate to \( 1/T \). Hence, this statement is incorrect.

Analyze statement (c)

Statement (c) proposes that a plot of \( \log K_p \) versus \( 1/T \) is linear. According to the Van't Hoff equation, \( \log K_p = -\Delta H/(RT) + \text{constant} \), indicating that this plot is linear. Thus, this statement is correct.

Analyze statement (d)

Statement (d) claims that a plot of \( \log[x] \) versus time is linear for a zero-order reaction. For zero-order reactions, the concentration \([x]\) decreases linearly over time, but \( \log[x] \) versus time would not be linear. Thus, this statement is incorrect.

Key concepts

Van't Hoff Equation

The Van't Hoff equation is a vital concept in understanding how equilibrium constants change with temperature. This can be particularly useful in predicting how reaction conditions will influence the position of equilibrium.

The equation is expressed as follows: \[ \log K_p = - \frac{\Delta H}{RT} + \text{constant} \]where:

• \(K_p\) is the equilibrium constant based on partial pressures.
• \(\Delta H\) is the enthalpy change of the reaction.
• \(R\) is the universal gas constant.
• \(T\) is the temperature in Kelvin.

This equation reveals that a plot of \(\log K_p\) versus \(1/T\) will form a straight line. The slope of the line is directly related to the enthalpy change, indicating whether the reaction is exothermic or endothermic. This relationship helps scientists understand how external temperature changes will affect chemical reactions.

In practical applications, this linear relationship is often used in calculations for reaction tuning and optimizing industrial chemical processes.

Zero-Order Reaction

Zero-order reactions are unique because the rate of reaction is constant and does not depend on the concentration of the reactants. In mathematical terms, if we look at the rate law for a zero-order reaction, it is expressed as:\[ \text{Rate} = k \]where:

• \(k\) is the rate constant.

In zero-order reactions, the concentration of the reactant decreases linearly over time. However, it’s important to note that a plot of \( \log[x] \) versus time will not be linear for such reactions. Instead, the relationship that displays linearity is a plot of \([x]\) versus time, which shows the linear decrease directly.

This constant rate is often observed in cases where a reactant is saturated on a surface, such as enzyme reactions where the enzyme is saturated with substrate. Understanding the zero-order kinetics is crucial in fields like pharmacology, where drug concentration can affect how they are metabolized in the body.

Ideal Gas Law

The ideal gas law is an essential equation in chemistry that provides a simple but powerful relationship between the pressure, volume, temperature, and the number of moles of an ideal gas. It is commonly stated as:\[ PV = nRT \]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 equation helps us understand how varying one of the parameters affects the others when dealing with an ideal gas. For example, when keeping the volume constant, the pressure of the gas should increase with rising temperature.

Understanding the ideal gas law is crucial for comprehending more complex gas laws, such as Boyle's Law and Charles's Law, which describe more specific relationships between the variables when certain conditions are held constant.