Question

Calculate the amount of heat required to completely sublime 50.0 \(\mathrm{g}\) of solid dry ice \(\left(\mathrm{CO}_{2}\right)\) at its sublimation temperature. The heat of sublimation for carbon dioxide is 32.3 \(\mathrm{k} / \mathrm{mol}\) .

Short answer

Heat required \(= \frac{50.0~g}{44.01~g/mol} \times 32.3 \frac{kJ}{mol} = 36.68~kJ\).

Step-by-step solution

Understand the concept of heat of sublimation

The heat of sublimation is the amount of energy required to change a substance from its solid phase to a gaseous phase without passing through a liquid phase. It is typically expressed in kilojoules per mole (kJ/mol).

Calculate the number of moles of \(\mathrm{CO}_2\)

First, determine the number of moles of \(\mathrm{CO}_2\) by using its molar mass. The molar mass of \(\mathrm{CO}_2\) is approximately 44.01 \(\mathrm{g/mol}\). Number of moles \(= \frac{mass}{molar~mass} = \frac{50.0~g}{44.01~g/mol}\).

Calculate the total heat required for sublimation

Now, multiply the number of moles of \(\mathrm{CO}_2\) by the heat of sublimation to find the total amount of heat required. Total heat \(Q = number~of~moles \times heat~of~sublimation\).

Key concepts

Thermochemistry

Thermochemistry is a branch of chemistry that deals with the study of heat energy associated with chemical reactions and physical transformations, such as phase transitions. In the context of our exercise, we are looking at the thermochemical process of sublimation, where solid carbon dioxide (also known as dry ice) directly changes into a gaseous state.

To understand this process thermochemically, we need to consider the heat of sublimation, which is the amount of energy needed to convert one mole of a solid directly into a gas at a constant temperature. This heat is absorbed from the surroundings, since sublimation is an endothermic process. Knowing the heat of sublimation, as given for carbon dioxide, allows us to calculate the energy required for a given mass of the substance to sublimate.

The implications of thermochemistry extend far beyond classroom exercises, as it also plays a crucial role in fields such as materials science, environmental science, and energy technology. Thermochemical data help us predict how substances will behave under different thermal conditions, which is essential for the design and operation of chemical processes.

Phase Transitions

Phase transitions are transformations between different states of matter: solid, liquid, and gas. When a substance undergoes a phase transition, it absorbs or releases energy. The process of sublimation, where a solid turns directly into a gas, is one of these phase transitions.

In our exercise, dry ice sublimes at its sublimation temperature, which means it's transitioning from a solid to a gas. This phase change requires a certain amount of energy, measured as the heat of sublimation. It's important to understand that during this transition, the temperature of the substance does not change; instead, the added energy goes into changing the state of the matter.

Phase transitions are not limited to sublimation; they also include melting, freezing, vaporization, condensation, and deposition. Each process has its own characteristic energy requirement or release, such as heat of fusion for melting and heat of vaporization for boiling. Understanding phase transitions is crucial for many aspects of science and engineering, including weather patterns, food preservation, and the production of electronic components.

Mole Concept

The mole concept is a fundamental principle in chemistry that allows us to count particles at the atomic and molecular levels by weighing macroscopic quantities of material. A mole is defined as the amount of substance that contains the same number of entities (atoms, molecules, ions, etc.) as there are atoms in 12 grams of carbon-12. This number is Avogadro's number, approximately equal to \(6.022 \times 10^{23}\) entities per mole.

In the exercise, we use the mole concept to convert the mass of dry ice into the number of moles, as thermochemical calculations are typically based on moles of substances. To perform this conversion, we divide the mass of the dry ice by its molar mass—that is, the mass of one mole of carbon dioxide molecules, which is approximately 44.01 grams per mole.

This step is crucial when linking macroscopic measurements, such as the weight of a sample, with microscopic properties, like the energy required to sublimate each molecule. The mole concept thus provides a bridge between the visible world and the atomic scale, enabling us to perform calculations that are essential in both theoretical and applied chemistry.