Problem 82 - Chemistry An Atoms First Approach

Question

Some water is placed in a coffee-cup calorimeter. When \(1.0 \mathrm{g}\) of an ionic solid is added, the temperature of the solution increases from \(21.5^{\circ} \mathrm{C}\) to \(24.2^{\circ} \mathrm{C}\) as the solid dissolves. For the dissolving process, what are the signs for \(\Delta S_{\mathrm{sys}}, \Delta S_{\text {surr, and }}\) \(\Delta S_{\text {univ }} ?\)

Step-by-Step Solution

Answer

The dissolving process of the ionic solid is exothermic. As the ionic solid dissolves, the entropy of the system increases (\(\Delta S_{\mathrm{sys}} > 0\)). The entropy of the surroundings also increases (\(\Delta S_{\mathrm{surr}} > 0\)) because heat is released to the surroundings. Therefore, the entropy change of the universe increases (\(\Delta S_{\mathrm{univ}} > 0\)). So, the signs for the entropy changes are: \(\Delta S_{\mathrm{sys}} > 0\), \(\Delta S_{\mathrm{surr}} > 0\), and \(\Delta S_{\mathrm{univ}} > 0\).

Step 1: Determine if the process is exothermic or endothermic

The temperature of the solution increases from \(21.5^{\circ} \mathrm{C}\) to \(24.2^{\circ} \mathrm{C}\) when the ionic solid dissolves. This means that heat is being released into the solution, as the temperature is rising. This is an exothermic process.

Step 2: Analyze ∆S of the system

Entropy is a measure of the number of possible microstates, which is usually associated with the "disorder" or "randomness" of a system. When the ionic solid dissolves in water, the ions become more dispersed in the solution, increasing the disorder of the system. Thus, the entropy of the system increases. So, \(\Delta S_{\mathrm{sys}} > 0\).

Step 3: Analyze ∆S of the surroundings

Since the process is exothermic, heat is released from the system to the surroundings, and the temperature of the surroundings (the coffee-cup calorimeter) increases. As the temperature increases, the random motions of the particles increase, leading to an increase in entropy. Therefore, \(\Delta S_{\mathrm{surr}}>0\).

Step 4: Analyze ∆S of the universe

The entropy change of the universe is the sum of the entropy changes of the system and the surroundings. In this case, both the system and the surroundings have positive entropy changes. So, when summing them, \(\Delta S_{\mathrm{univ}} = \Delta S_{\mathrm{sys}} + \Delta S_{\mathrm{surr}} > 0\). In conclusion, the signs for the entropy changes are as follows: \(\Delta S_{\mathrm{sys}} > 0\), \(\Delta S_{\mathrm{surr}} > 0\), and \(\Delta S_{\mathrm{univ}} > 0\).

Key Concepts

Exothermic process

An exothermic process is a chemical reaction or physical change that releases heat to its surroundings. This heat release results in an increase in temperature, which can be measured using calorimetry. When a substance dissolves in water exothermically, the energy stored in the substance's bonds is released more than the energy absorbed to break the solvent's bonds.

During this type of process, the energy flow exits the system, which is the substance dissolving, and enters the surroundings, which can include the solvent and the container holding the solution. This energy transfer is commonly observed in everyday phenomena like the warming of a solution when salt dissolves in water, or the heat emitted from hand warmers that rely on exothermic reactions.

Entropy

Entropy, symbolized by the letter S, is a thermodynamic quantity representing the amount of disorder or randomness in a system. The second law of thermodynamics states that the total entropy of an isolated system can never decrease over time. Entropy is often viewed as a measure of molecular uncertainty, and the change in entropy, represented by \(\Delta S\), indicates a system's energy dispersal.

In the context of dissolving, an ionic solid disperses into its constituent ions when it dissolves in water, leading to an increased number of microstates and higher disorder. Therefore, dissolving typically increases the entropy of the system (\(\Delta S_{\mathrm{sys}} > 0\)). An increased entropy suggests a more stable and favorable state.

Calorimetry

Calorimetry is the science of measuring the heat of chemical reactions or physical changes. A calorimeter is an instrument that measures the amount of heat involved in such processes. In a coffee-cup calorimeter, like in our dissolving example, the heat change experienced by the water can act as an indicator of the heat absorbed or released by the reaction. The total heat content or enthalpy change (\(\Delta H\)) is intricately linked to both entropy change (\(\Delta S\)) and the overall spontaneity of a process in thermodynamics.

The temperature change in the calorimeter provides valuable information on whether the process is endothermic (absorbing heat) or exothermic (releasing heat), as well as the sign of the enthalpy change. By calculating the heat using temperature change and the known heat capacity, one can understand the energy dynamics of the dissolving process.

Thermodynamics

Thermodynamics is a branch of physics concerned with heat and temperature and their relation to energy and work. The laws of thermodynamics govern the principles of energy transfer within a chemical reaction or physical change. When a substance dissolves, these laws can be used to predict whether the process is spontaneous based on changes in enthalpy (\(\Delta H\)), entropy (\(\Delta S\)), and temperature.

The dissolving process is affected by the thermodynamic concept of Gibbs free energy (\(\Delta G\)), which combines enthalpy and entropy changes. A negative \(\Delta G\) means the process is spontaneous, and positive \(\Delta G\) indicates non-spontaneity. When both \(\Delta S_{\mathrm{sys}}\) and \(\Delta S_{\mathrm{surr}}\) are positive, as in the example exercise, it is highly probable that \(\Delta G\) will be negative, favoring the dissolving process under the given conditions.