Question

Van't Hoff equation is

• A. \(\frac{\partial x}{\partial T} = kC^n\)
• B. \(\frac{\partial x}{\partial t} = -kC^n\)
• C. \(\frac{\partial x}{\partial T} = kC\)
• D. \(\frac{\partial x}{\partial t} = kC\)

Hint:

Be aware that questions in exams can sometimes be flawed. If you encounter a mismatch between the question and the options, first recall the correct definition/formula (here, the Van't Hoff equation). Then, analyze the options to see what concept they might relate to (here, chemical kinetics). Choose the most plausible option based on the related concept.

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
This question is likely flawed. The question asks for the Van't Hoff equation, but the options provided are expressions related to chemical reaction rates (kinetics), not the Van't Hoff equation. The Van't Hoff equation describes the relationship between the equilibrium constant (\(K\)) of a reaction and the temperature (\(T\)). However, we must choose the best fit or intended answer from the given options. The selected answer is a general form of a rate law.

Step 2: Key Formula or Approach:
The actual Van't Hoff equation is: \[ \frac{d(\ln K)}{dT} = \frac{\Delta H^\circ}{RT^2} \] where \(K\) is the equilibrium constant, \(T\) is the absolute temperature, \(R\) is the ideal gas constant, and \(\Delta H^\circ\) is the standard enthalpy change of the reaction.
The options relate to the rate law, which is generally expressed as: \[ \text{Rate} = k[A]^m[B]^n... \] If we let \(x\) or \(C\) be the concentration of a reactant, the rate of its consumption is \(-\frac{dC}{dt}\). So, a general rate law can be written as \(-\frac{dC}{dt} = kC^n\), where \(n\) is the order of the reaction.

Step 3: Detailed Explanation:
Let's analyze the discrepancy and the given options.

The Question: It explicitly asks for the "Van't Hoff equation". As stated above, none of the options represent the true Van't Hoff equation.

The Options: The options are differential equations involving concentration (\(C\) or \(x\)), time (\(t\)), temperature (\(T\)), and a rate constant (\(k\)). These are forms of rate equations in chemical kinetics.

• (A) \(\frac{\partial x}{\partial T} = kC^n\): Relates change in concentration with temperature, which is incorrect for a rate law.

• (B) \(\frac{\partial x}{\partial t} = -kC^n\): This represents the rate of change of concentration of a reactant (\(x\) or \(C\)) with respect to time (\(t\)). The negative sign indicates that the concentration of the reactant is decreasing over time. 'n' is the order of the reaction. This is a valid general form for a rate law. The partial derivative symbol might be used loosely for a total derivative.

• (C) \(\frac{\partial x}{\partial T} = kC\): Incorrect, relates concentration change to temperature.

• (D) \(\frac{\partial x}{\partial t} = kC\): This is a rate law for a first-order reaction (\(n=1\)), but it represents the rate of formation of a product (positive sign).

Conclusion based on the checkmark: The checkmark is on option (B). Assuming the question intended to ask for a general rate law for the consumption of a reactant, option (B) is the most appropriate choice. It correctly shows that the rate of change of reactant concentration (\(\frac{\partial x}{\partial t}\)) is proportional to the concentration raised to some power (\(C^n\)), and the negative sign indicates consumption.

Step 4: Final Answer:
Despite the question incorrectly asking for the Van't Hoff equation, the selected answer, \(\frac{\partial x}{\partial t} = -kC^n\), represents a general rate law for the consumption of a reactant in a chemical reaction of order \(n\).