Problem 2 - Advanced Problems in Physical Chemistry for Competitive Examinations

Question

\(2 \mathrm{MnO}_{4}^{-}+16 \mathrm{H}^{+}+10 \mathrm{Cl}^{-} \rightarrow 2 \mathrm{Mn}^{2+}\) \(+5 \mathrm{Cl}_{2}(\mathrm{~g})+8 \mathrm{H}_{2} \mathrm{O}\) Above reaction is endothermic and hence the actual temperature of the reaction vessel (isolated from the surrounding) may be different from that expected. Given that the initial temperature of the reaction vessel was used in the calculations, how would, this affect the predicted value of moles of \(\mathrm{Cl}_{2}(n)\) according to equation: \(n=P V / R T\) (a) It would be greater than the actual value (b) It would be less than the actual value (c) It would be the same as the actual value (d) This cannot be determined from the information given

Step-by-Step Solution

Answer

It would be less than the actual value

Step 1: Understand the scenario and the equation

The given reaction is an endothermic process, meaning it absorbs heat. If the temperature in the reaction vessel increases, it indicates that heat has been absorbed from the reaction itself. The ideal gas law (\( n = \frac{PV}{RT} \) is used for calculating the moles of a gas, where P is pressure, V is volume, R is the ideal gas constant, and T is temperature in Kelvin.

Step 2: Analyze the effect of temperature on the ideal gas law

In the ideal gas equation, if the temperature (T) increases while pressure (P), volume (V), and the ideal gas constant (R) remain constant, the number of moles (n) of gas calculated using the initial temperature would be lower than the actual number of moles present because the value of T in the denominator has increased.

Step 3: Conclude the effect on predicted moles of Chlorine gas

Since the reaction is endothermic and the actual temperature of the reaction vessel would be higher, using the initial temperature in the ideal gas law calculation would result in a predicted value of moles of chlorine gas () that is less than the actual value.

Key Concepts

Understanding Endothermic Reactions

An endothermic reaction is a type of chemical process that absorbs heat from its surroundings, leading to a decrease in the temperature environment unless the system is isolated. In an endothermic reaction like the decomposition of manganese(IV) oxide, the energy required to break the existing bonds is greater than the energy released by forming new bonds. This energy difference is why external heat is needed for the reaction to proceed.

In the context of the equation provided, the temperature of the vessel may increase if the system is not perfectly isolated, a situation often neglected when applying the ideal gas law in calculations. This rise in temperature has a direct effect on the predicted value of moles of a gas. Since the student is expected to use the initial temperature rather than the actual temperature for such calculations, they should always consider the potential discrepancies in their results due to the endothermic nature of the reaction.

It's essential to grasp this concept for understanding how temperature changes affect chemical equilibria and reaction rates, which play a critical role in physical chemistry, especially when prepping for competitive exams.

Moles Calculation and the Ideal Gas Law

The calculation of moles of a substance, often represented by the symbol 'n', is a foundational concept in chemistry. The mole is a unit that allows chemists to convert between the mass of a substance and the number of its particles, atoms, ions, or molecules. When dealing with gases, the Ideal Gas Law (IGL) provides a useful way to calculate moles. The IGL equation is expressed as \( n = \frac{PV}{RT} \), where P represents pressure, V is volume, T is the absolute temperature in Kelvin, and R is the ideal gas constant.

An accurate moles calculation is crucial when predicting the stoichiometry of reactions. However, in a scenario involving endothermic reactions, temperature changes can lead to errors if the initial temperature, rather than the actual temperature, is used. These calculations become especially vital in laboratory settings and for students tackling physical chemistry competitive exams where precision is important.

Students should remember that the actual number of moles of gas present in a system at equilibrium can be different from the calculated value if the temperature is not correctly accounted for, due to the dependency of temperature in the IGL equation.

Competitive exams in physical chemistry often test a student's understanding of complex concepts like endothermic reactions and the calculations associated with the ideal gas law. Success in these exams requires a deep comprehension of not only the theoretical aspects but also the ability to solve practical problems accurately.

Factors that affect the outcomes of chemical reactions—temperature, pressure, volume, and moles—are frequently featured in exam questions. Students must be well-versed in applying the ideal gas law under different conditions, as this tends to be a favorite topic for examiners. Understanding the nature of endothermic reactions, and how they can impact the predicted and actual outcomes, enhances one's ability to provide correct answers during exams.

Exam preparation should include practice with a variety of problems, learning to quickly identify whether reactions are endothermic or exothermic, and how to effectively apply the ideal gas law. Mastering these skills can greatly increase the likelihood of achieving a top score on competitive physical chemistry exams.