Problem 11 - Chemistry The Science in Context

Question

Ice cubes melt in a glass of lemonade, cooling the lemonade from \(10.0^{\circ} \mathrm{C}\) to \(0.0^{\circ} \mathrm{C} .\) If the ice cubes are the system, what are the signs of \(\Delta S_{\text {sys }}\) and \(\Delta S_{\text {surr }} ?\)

Step-by-Step Solution

Answer

Question: Determine the signs of the entropy changes for the system (ice cubes) and surroundings (lemonade) when ice cubes melt in lemonade. Answer: The sign of the entropy change for the system (ice cubes) is positive (\(\Delta S_{\text {sys }}>0\)), while the sign of the entropy change for the surroundings (lemonade) is negative (\(\Delta S_{\text {surr }}<0\)).

Step 1: Identify the system and surroundings

In this exercise, the system consists of the ice cubes, while the surroundings are the lemonade in the glass. We will analyze the changes in entropy for both the system and the surroundings as heat is transferred between them.

Step 2: Determine the heat flow direction

Heat will flow from the warmer substance to the cooler one. In this case, since the lemonade is initially at \(10.0^{\circ}\mathrm{C}\) and the ice cubes are likely to be at \(0.0^{\circ}\mathrm{C}\), heat will flow from the lemonade to the ice cubes.

Step 3: Analyze the entropy change in the system (ice cubes)

As heat is transferred from the surroundings to the ice cubes, their temperature will increase, leading to an increase in the randomness of the particles. As a result, the system's entropy will increase. Mathematically, entropy change (ΔS) is positive when heat is absorbed by the system. Therefore, \(\Delta S_{\text {sys }}>0\).

Step 4: Analyze the entropy change in the surroundings (lemonade)

The heat lost by the lemonade in the process of melting the ice cubes will decrease its temperature, and its particles will become more ordered. The entropy of the surroundings (lemonade) will decrease as a result. Mathematically, entropy change (ΔS) is negative when heat is released by the system (in this case, the surroundings are releasing heat). Therefore, \(\Delta S_{\text {surr }}<0\).

Step 5: Report the signs of the entropy changes

In conclusion, \(\Delta S_{\text {sys }}>0\), which means the entropy of the system (ice cubes) increases, and \(\Delta S_{\text {surr }}<0\), which means the entropy of the surroundings (lemonade) decreases.

Key Concepts

Thermodynamics

Thermodynamics is a branch of physics that deals with heat, work, and energy. In our daily lives, we experience examples of thermodynamics through processes like boiling water or melting ice. The principles of thermodynamics are based on laws, most notably the first and second laws. The first law, also known as the law of energy conservation, states that energy cannot be created or destroyed, only transformed from one form into another. This is why when ice cubes melt in lemonade, the heat from the lemonade is not lost but transferred to the ice, causing it to melt.

The second law of thermodynamics introduces the concept of entropy, a measure of disorder or randomness in a system. It states that in an isolated system, entropy can never decrease over time. As heat is transferred, the second law predicts that the total entropy of the system plus the surroundings will either increase or remain constant. For the melting ice cubes scenario, the system (ice cubes) absorbs heat, leading to the increase of its entropy while the surroundings (lemonade) lose heat, hence their entropy decreases.

Heat Transfer

Heat transfer is the movement of thermal energy from one object or substance to another. It plays a central role in thermodynamics and occurs in three main ways: conduction, convection, and radiation. In the case of ice melting in lemonade, conduction is the primary way heat is transferred. Conduction is the transfer of heat between substances that are in direct contact with each other.

As the warmer lemonade is in direct contact with the cooler ice cubes, thermal energy is transferred from the lemonade to the ice, hence the ice melts. This process will continue until thermal equilibrium is reached, meaning the lemonade and the ice cubes will eventually settle at the same temperature. Understanding heat transfer helps us to realize why the lemonade cools down as the ice melts – it's the result of heat leaving the lemonade and entering the ice cubes.

State Functions

In thermodynamics, state functions are properties of a system that depend only on its current state, not on the path the system took to reach that state. Entropy is a key example of a state function. It doesn't matter how the heat was transferred or what process occurred; the change in entropy will always be the same for a given initial and final state of a system.

The value of state functions like entropy helps to describe the system in terms of equilibrium, phase transitions, and predict whether a process is spontaneous or nonspontaneous. In the melting ice example, the entropy change for the ice and lemonade can be calculated based on their beginning and final states, such as their temperatures, ignoring the actual process of the ice melting or the lemonade cooling.