Problem 34 - Chemistry The Science in Context

Question

Another decorative "ice" sculpture is carved from dry ice (solid \(\mathrm{CO}_{2}\) ) and held at its sublimation point of \(-78.5^{\circ} \mathrm{C} .\) What is the entropy change to the universe when the \(\mathrm{CO}_{2}\) sculpture, weighing \(389 \mathrm{g},\) sublimes on a granite tabletop if the temperature of the granite is \(12^{\circ} \mathrm{C}\) and the process occurs reversibly? Assume a final temperature for the \(\mathrm{CO}_{2}\) vapor of \(-78.5^{\circ} \mathrm{C} .\) The heat of sublimation of \(\mathrm{CO}_{2}\) is \(26.1 \mathrm{kJ} / \mathrm{mol}\).

Step-by-Step Solution

Answer

Answer: The total entropy change in the universe is 0.376 kJ/K.

Step 1: Convert the given mass of CO₂ to moles.

To do this, we will divide the given mass by the molar mass of CO₂. Molar mass of CO₂=44.01 g/mol Moles of CO₂= Mass of CO₂ / Molar mass of CO₂ Moles of CO₂= 389 g / 44.01 g/mol=8.84 mol

Step 2: Calculate the heat absorbed by CO₂.

To do this, we will multiply the number of moles by the heat of sublimation. Heat absorbed by CO₂ = moles of CO₂ × heat of sublimation of CO₂ Heat absorbed by CO₂ = 8.84 mol × 26.1 kJ/mol = 230.6 kJ

Step 3: Calculate the entropy change of CO₂ during sublimation

To do this, we will divide the heat absorbed by CO₂ by the absolute temperature of dry ice: -78.5°C (or 194.65 K using the formula: K = °C + 273.15) Entropy change of CO₂ = Heat absorbed by CO₂ / Temperature of sublimation point Entropy change of CO₂ = 230.6 kJ / 194.65 K = 1.185 kJ/K

Step 4: Calculate the heat released by the granite tabletop.

Since the process is reversible, the heat absorbed by the CO₂ must be equal to the heat lost by the granite tabletop, so: Heat released by granite = Heat absorbed by CO₂ = 230.6 kJ

Step 5: Calculate the entropy change of the granite tabletop.

To do this, we will divide the heat released by the granite by the absolute temperature of the granite tabletop: 12°C (or 285.15 K using the formula: K = °C + 273.15) Entropy change of granite = Heat released by granite / Temperature of granite Entropy change of granite = -230.6 kJ / 285.15 K = -0.809 kJ/K

Step 6: Calculate the total entropy change in the universe.

To do this, we will add the entropy changes of CO₂ and the granite tabletop. Total entropy change in the universe = Entropy change of CO₂ + Entropy change of granite Total entropy change in the universe = 1.185 kJ/K + (-0.809 kJ/K) = 0.376 kJ/K

Key Concepts

Chemical Thermodynamics

Chemical thermodynamics is the study of energy changes during chemical reactions and processes, such as sublimation in this exercise. It focuses on understanding concepts like heat transfer, work done, and changes in properties like entropy and enthalpy.
In the world of chemical reactions and transformations, thermodynamics helps us predict if a process can occur spontaneously, and if so, in which direction. One of the key principles involved in these predictions is the second law of thermodynamics, which involves the concept of entropy.
Entropy, denoted by the symbol \( S \), is often described as a measure of a system's randomness or disorder. When a process occurs, like sublimation, observing the entropy change provides insight into the overall direction and feasibility of the process.

Heat Sublimation

Heat sublimation is an important concept when dealing with phase changes, particularly for substances transitioning directly from solid to gas without passing through a liquid phase. For instance, dry ice sublimation involves carbon dioxide changing directly from solid to gas.
The heat of sublimation is the amount of heat required to convert one mole of a solid substance directly into its gaseous form, without increasing its temperature. For carbon dioxide in this example, the heat of sublimation is given as 26.1 kJ/mol. This value represents the energy needed to break intermolecular forces holding the solid molecules in the dry ice structure.
When dry ice sublimes on a granite tabletop, it absorbs heat equal to its heat of sublimation, which in turn affects the entropy of both the carbon dioxide and its environment, as calculated.

Reversible Process

A reversible process is an idealized concept in thermodynamics where a system transitions from one state to another perfectly and can return to the original state without any net change in the system and surroundings. In real-world applications, true reversibility is impossible, but it provides a useful model.
In the context of the dry ice sublimation exercise, the assumption that the process is reversible allows us to equate the heat absorbed by carbon dioxide to the heat released by the granite surface. Consequently, changes in entropy can be calculated with precision.
The meticulous balance maintained in reversible processes also implies that the change in total entropy of the universe remains minimized. However, in any practical process, there will always be an increase in the entropy of the universe, highlighting the irreversible nature.

Carbon Dioxide Sublimation

Carbon dioxide sublimation is the transition of carbon dioxide from its solid form, known as dry ice, directly to a gas at atmospheric pressure and specific conditions, without ever becoming liquid. This phenomenon is utilized in various applications due to the cooling effects of sublimating dry ice.
In the scenario of the granite tabletop and dry ice sculpture, the transition occurs at a constant temperature of \(-78.5^{\circ}\mathrm{C}\), which is the sublimation point of CO₂. Observing the sublimation of CO₂ provides valuable insights into energy transfer and entropy changes.
When carbon dioxide sublimes, it absorbs energy from its surroundings, which typically causes a temperature drop in the immediate vicinity, like the granite surface. Understanding these transitions helps address larger questions about phase changes and energy dynamics in thermodynamic systems.