Question

The molar heat capacities of \(\mathrm{A}, \mathrm{B}\) and \(\mathrm{C}\) are in the ratio \(1: 2: 3 .\) The enthalpy change for the reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}\) at temperature \(T_{1}\) is \(\Delta H_{1} .\) Assuming that the heat capacities do not change with temperature, the enthalpy change, \(\Delta H_{2}\), at temperature, \(T_{2}\left(T_{2}>T_{1}\right)\) will be (a) greater than \(\Delta H_{1}\) (b) equal to \(\Delta H_{1}\) (c) less than \(\Delta H_{1}\) (d) greater or less than \(\Delta H_{1}\), depending on the values of \(T_{2}\) and \(T_{1}\).

Short answer

The enthalpy change, \(\Delta H_{2}\), at temperature \(T_{2}\) will be (b) equal to \(\Delta H_{1}\) since the heat capacities do not change with temperature and hence the enthalpy change is independent of temperature.

Step-by-step solution

Understand the Relationship of Molar Heat Capacities

The molar heat capacities of substances A, B and C determine how much heat is required to raise the temperature of one mole of each substance by one degree Celsius. In this case, their heat capacities are in the ratio of 1:2:3. This implies that C, having the largest heat capacity, would absorb more heat for the same temperature change compared to A or B.

Apply Hess's Law

According to Hess's Law, the total enthalpy change of a reaction is the same, no matter how many steps the reaction is carried out in, given that the initial and final conditions are the same. Since the heat capacities are constant and we are dealing with a direct conversion from A+B to C, the enthalpy change should theoretically be independent of the temperature.

Consider the Effect of Temperature on Enthalpy

If the heat capacities do not change with temperature, then the amount of heat absorbed or released per degree temperature change remains constant. Therefore, the enthalpy change at different temperatures would remain the same because the substances absorb or release the same amount of heat per degree temperature change irrespective of the starting temperature.

Key concepts

Molar Heat Capacity

The term molar heat capacity refers to the amount of heat energy required to raise the temperature of one mole of a substance by one degree Celsius (or one Kelvin). In chemical reactions, understanding molar heat capacity is crucial because it influences how substances in a reaction will react to heat and can affect the overall energy change during the process.

For example, in the exercise mentioned, substances A, B, and C have molar heat capacities in the ratio of 1:2:3. This means that if all three substances are exposed to the same amount of heat, C would exhibit the greatest temperature change, followed by B, then A. If the molar heat capacities remain constant—which is a key assumption here—the enthalpy change of the reaction should not depend on the temperature since the same amount of heat would cause the same temperature change at any starting temperature.

Hess's Law

Hess's Law states that the total enthalpy change for a chemical reaction is the same, no matter how the reaction occurs, as long as the initial and final conditions are the same. This principle is a cornerstone in chemical thermodynamics because it allows for the calculation of enthalpy changes for reactions that are difficult to measure directly.

Considering the exercise provided, Hess's Law implies that regardless of the temperatures involved, the enthalpy change from A+B to C should theoretically remain constant. This is because the initial state (A+B) and final state (C) of the reaction are the same, regardless of whether the reaction takes place at temperature T1 or T2.

The reason temperature doesn't affect the enthalpy change in this ideal scenario is that we assume the molar heat capacities are constant. However, in real-life scenarios, where molar heat capacities can vary with temperature, Hess's Law still helps us calculate the enthalpy change by considering the heat capacities at different temperatures.

Chemical Thermodynamics

Chemical thermodynamics is the branch of physical chemistry that deals with the energy and work aspects of chemical and physical transformations. It's mainly concerned with heat and temperature and their relation to energy and work.

• Energy can be exchanged between physical systems as heat or work.
• The laws of thermodynamics govern how and why energy is transferred.
• Understanding spontaneous processes and equilibria is critical for predicting the behavior of chemical reactions.

Within the context of our exercise, chemical thermodynamics tells us that if heat capacities were to change with temperature, this would complicate the calculation of enthalpy changes. However, by assuming constant heat capacities, as done in this exercise, thermodynamics simplifies the problem, leading to the conclusion that enthalpy changes during a reaction are temperature independent under these conditions.

In essence, the intricate dance of energy exchange during reactions, is dictated by the rigorous laws of thermodynamics, offering students and scientists a framework to understand and predict outcomes of chemical processes.