Question

When a mole of dry ice, \(\mathrm{CO}_{2}(s)\), is converted to \(\mathrm{CO}_{2}(g)\) at atmospheric pressure and \(-78^{\circ} \mathrm{C}\), the heat absorbed by the system exceeds the increase in internal energy of the \(\mathrm{CO}_{2}\). Why is this so? What happens to the remaining energy?

Short answer

During the sublimation process of dry ice (CO₂) at atmospheric pressure and -78°C, the heat absorbed by the system (q) is greater than the increase in internal energy (∆U) because some of the absorbed heat is used to perform work (w) as the CO₂ gas expands under atmospheric pressure. According to the First Law of Thermodynamics, ∆U = q - w. As a result, the remaining energy is utilized to perform mechanical work, displacing the surrounding atmosphere to accommodate the expanding gaseous CO₂, and is transferred to the surroundings as work.

Step-by-step solution

Understand the Sublimation Process

Sublimation is the process in which a substance changes from a solid to a gaseous state, without passing through the liquid phase. In this case, dry ice, CO₂(s), is transformed directly into CO₂(g) at atmospheric pressure and -78°C.

Assess the Heat and Internal Energy Relationship

During the sublimation process, the CO₂(s) absorbs heat from its surroundings. According to the First Law of Thermodynamics, the change in internal energy (∆U) of a system is equal to the heat absorbed (q) minus the work done by the system (w): \[∆U = q - w\]

Determine the Work Done in the System

As the dry ice, CO₂(s), converts into CO₂(g) at atmospheric pressure, the gaseous CO₂ will expand, and the system will perform work. The work done by the system is given by the equation: \[w = P × ∆V\] where P is atmospheric pressure, and ∆V is the change in volume of CO₂.

Explain the Excess Heat Absorbed by the System

From the First Law of Thermodynamics equation, we know that: \[∆U = q - w\] Since the dry ice (CO₂) is performing work during the sublimation process (expanding under atmospheric pressure), some of the heat absorbed (q) is used to perform this work (w). Therefore, the increase in internal energy (∆U) of the CO₂ is less than the total heat absorbed, as some energy is used for the work done by the system.

Identify the Destination of the Remaining Energy

The remaining energy represents the work done by the system during the sublimation process. As the CO₂ gas expands under atmospheric pressure, this energy is utilized to perform mechanical work to displace the surrounding atmosphere and make room for the expanding gaseous CO₂. Thus, the remaining energy is transferred to the surroundings as work.

Key concepts

Dry Ice - Exploring Its Sublimation

Dry ice is the solid form of carbon dioxide ( CO₂). It is called 'dry' because unlike regular ice, it doesn't melt into a liquid but instead sublimates directly into its gaseous form. This occurs at atmospheric pressure and at a very low temperature of -78°C. This peculiar property makes dry ice a popular choice for keeping goods cold, especially during transportation.

• Sublimation: Dry ice ( CO₂(s)) skips the liquid phase, going straight from a solid to a gas.
• No liquid residue: This leaves no wet residue behind.

This transformation occurs because the molecules in the solid phase receive enough energy to break free from their rigid structure, becoming gas molecules that float in the air. When you witness dry ice sublimating, you often see a foggy cloud; however, that cloud is not CO₂ gas but rather water vapor condensing in the cold air around the sublimating dry ice.

Understanding Internal Energy

Internal energy refers to the sum total of all kinetic and potential energies of the molecules within a system. When dry ice sublimates, the CO₂ molecules absorb heat from their surroundings. This absorbed heat increases the molecules' kinetic energy, causing them to move more freely as they transition into the gaseous state.
According to the First Law of Thermodynamics, the change in internal energy (∆U) of the system is determined by the balance between the heat absorbed (q) and the work done by the system (w):\[∆U = q - w\]When dry ice sublimates, not all the absorbed heat contributes to increasing internal energy, which means that ∆U is less than q due to work done during the expansion, which we'll explore next.

• Absorption of Heat: Heat (q) flows into the system, raising the energy level of the molecules.
• Energy Distribution: Part of this heat is used for increasing internal energy, while the rest is used for work done.

Work Done During Expansion

As dry ice undergoes sublimation at atmospheric pressure, it occupies more space as it transitions to gaseous CO₂. The expanding gas performs work on its surroundings by pushing against atmospheric pressure, causing displacement. This work accounts for the difference between the heat absorbed and the change in internal energy.
The work done (w) is calculated as:\[w = P \times ∆V\]where P is the atmospheric pressure and ∆V is the change in volume as the CO₂ gas expands.

• Performing Work: Energy is spent in doing work by expanding the gas.
• Energy Transfer: This process transfers the remaining energy to the surroundings.

Understanding this concept illustrates why not all absorbed heat raises internal energy. Instead, part of it facilitates the expansion against atmospheric pressure, performing mechanical work.