Problem 11 - Chemistry: Introducing Inorganic, Organic and Physical Chemistry

Question

An ice cube of mass \(18 \mathrm{g}\) is added to a large glass of water just above \(0^{\circ} \mathrm{C}\). Calculate the change of entropy for the ice and for the water (without the ice). (Section 14.4 ) (The enthalpy change of fusion for water is \(+6.01 \mathrm{kJ} \mathrm{mol}^{-1}\).)

Step-by-Step Solution

Answer

The change of entropy for the ice is \( +0.0220 \, \text{kJ/K} \), and for the water is \( -0.0220 \, \text{kJ/K} \).

Step 1: Convert mass to moles

First, convert the mass of the ice cube from grams to moles. The molar mass of water (H2O) is approximately 18 g/mol. Hence, the ice cube's mass of 18 g corresponds to \( \frac{18 \, \text{g}}{18 \, \text{g/mol}} = 1 \, \text{mol}\) of water.

Step 2: Determine the heat absorbed by the ice

The heat absorbed by the ice as it melts is calculated using the enthalpy change of fusion. Since we have 1 mole of ice, the heat absorbed \( q \) during melting is given by \( q = n \times \Delta H_{\text{fusion}} = 1 \, \text{mol} \times 6.01 \, \text{kJ/mol} = 6.01 \, \text{kJ} \).

Step 3: Calculate change in entropy for the ice

The change in entropy \( \Delta S_{\text{ice}} \) for the ice as it melts can be calculated using the formula \( \Delta S = \frac{q}{T} \). The temperature \( T \) is in Kelvin, so we convert \( 0^{\circ} \text{C} \) to \( 273.15 \text{K} \). Thus, \( \Delta S_{\text{ice}} = \frac{6.01 \, \text{kJ}}{273.15 \, \text{K}} = 0.0220 \, \text{kJ/K} \).

Step 4: Calculate change in entropy for the water

The glass of water releases the same amount of heat that the ice absorbs, which is \( 6.01 \, \text{kJ} \). Therefore, the change in entropy \( \Delta S_{\text{water}} \) for the water is given by \( \Delta S_{\text{water}} = -\frac{6.01 \, \text{kJ}}{273.15 \, \text{K}} = -0.0220 \, \text{kJ/K} \). The negative sign indicates that the water is losing entropy.

Key Concepts

Entropy

Entropy is a measure of the disorder or randomness in a system. It is a central concept in thermodynamics, representing the irreversibility of natural processes. In simple terms, entropy can be thought of as the degree of disorder or the amount of chaos in a system.
- When ice melts, the orderly structure of the solid transitions to a more disordered liquid form. This increases entropy. - The corresponding decrease in entropy for the water indicates a reduction in the system's randomness as the water loses heat.
In our exercise, we observe that as ice melts, its entropy increases by 0.0220 kJ/K. Meanwhile, the water's entropy decreases by the same magnitude but negatively, illustrating the principle of conservation of energy, where the total change in entropy is a balance between gained and lost disorder.

Enthalpy Change of Fusion

The enthalpy change of fusion (8.6.01 28.kJ/mol929 represents the amount of energy required to convert one mole of a solid into a liquid at its melting point. This energy is crucial for overcoming the forces holding the solid's structure together.
- For water, the enthalpy change signifies how much heat is needed to transform ice to water without changing temperature. - This energy does not raise the temperature of the substance; instead, it breaks the bonds, allowing the phase change.
In this context, when 1 mole of ice absorbs 6.01 kJ of energy, it undergoes a phase change from solid to liquid, indicating the role of the enthalpy change of fusion in driving phase transitions.

Phase Change

A phase change occurs when a substance transitions from one state of matter to another: solid, liquid, or gas. During a phase change, the temperature of a substance remains constant until it completely changes state.
- This means that during the melting of ice, while it absorbs heat, the temperature remains at 0°C until all the ice has melted. - The energy supplied during this phase change doesn't increase kinetic energy or temperature but is used to disrupt the molecular bonds in the solid phase.
In our exercise, the phase change is the transition of ice to liquid water. This process involves an energy transfer without temperature change, which is pivotal in executing the concept of entropy adjustments as seen in the calculations provided.

Heat Transfer

Heat transfer refers to the energy exchanged between physical systems due to a temperature difference. It can occur through conduction, convection, or radiation. In this exercise, heat transfer is critical in understanding how ice melts when exposed to temperatures above 0°C.
- Here, conduction is the predominant mechanism. The warmer water transfers heat to the cooler ice, causing it to melt. - The water loses 6.01 kJ of heat, which the ice gains directly, signified by the positive and negative signs in our entropy calculations.
This balance in heat transfer illustrates the principle of energy conservation, where energy lost by the surroundings is equal to energy gained by the system, facilitating a uniform understanding of the thermodynamic behavior underlining ice melting.