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

When a mole of dry ice is converted to gaseous CO2 at atmospheric pressure and -78 °C, the heat absorbed by the system exceeds the increase in internal energy because some of the energy is used to do work on the atmosphere as the CO2(g) expands. The remaining energy goes into pushing the surrounding atmospheric gases outward due to the increased volume, effectively dispersing the remaining energy into the environment. This can be explained using the first law of thermodynamics: $\mathrm{\Delta }U=q-W$, where $\mathrm{\Delta }U$ is the change in internal energy, $q$ is the heat absorbed by the system, and $W$ is the work done by the system.

Step-by-step solution

Recall the first law of thermodynamics

The first law of thermodynamics states that the change in internal energy of a system is equal to the heat added to the system minus the work done by the system: $$\mathrm{\Delta }U=q-W$$ where $\mathrm{\Delta }U$ is the change in internal energy, $q$ is the heat absorbed by the system, and $W$ is the work done by the system.

Identify the given information

We are given that: - A mole of dry ice, CO2(s), is converted to CO2(g) at atmospheric pressure and -78 °C. - The heat absorbed by the system (q) is greater than the increase in internal energy of the CO2 ($\mathrm{\Delta }U$).

Explain why the heat absorbed exceeds the change in internal energy

Since the heat absorbed (q) is greater than the increase in internal energy ($\mathrm{\Delta }U$), we can say that: $$q>\mathrm{\Delta }U$$ From the first law of thermodynamics, we know that: $$\mathrm{\Delta }U=q-W$$ Combining the two, we get: $$q-W>\mathrm{\Delta }U$$ $$W>q-\mathrm{\Delta }U$$ This means that the work done by the system is greater than the difference between the heat absorbed and the increase in internal energy. When dry ice sublimates, it expands in volume, so the CO2(g) does work on the atmosphere by pushing it out of the way. This work is positive, meaning the system loses energy when doing work. Therefore, the energy that isn't going towards increasing the internal energy of the CO2 goes into doing work on the surroundings.

Determine what happens to the remaining energy

The remaining energy that isn't going towards increasing the internal energy of the CO2 goes into doing work on the atmosphere. As the CO2(g) expands, it pushes the surrounding atmospheric gases outward due to the increased volume. This energy is transferred to the surrounding atmosphere, effectively dispersing the remaining energy into the environment.

Key concepts

Understanding Thermodynamic Systems

A thermodynamic system is essentially a concept used in physics and chemistry to describe a controlled portion of the universe that we are interested in studying. Simply put, it's a defined space where we can observe and measure energy changes, such as heat and work. In the given exercise, the conversion of dry ice to gas at a constant temperature is a classic example of a thermodynamic process.

The science of thermodynamics divides systems into three types: closed, open, and isolated. A closed system can exchange energy (in the form of heat or work) but not matter with its surroundings, while an open system can exchange both energy and matter. In contrast, an isolated system does not exchange either energy or matter with the environment. The sublimation of dry ice occurs in a closed system where the CO2 can transition from a solid to a gaseous state without mass leaving the system, but energy in the form of heat is being transferred.

Enthalpy and Sublimation

When we delve into the specifics of thermodynamics, enthalpy is a term that often comes up. It refers to the total heat content of a system and is a measure of the energy needed for a substance to undergo a transformation at constant pressure – for instance, when a solid turns directly into a gas, a process known as sublimation.

The sublimation of dry ice involves a significant energy change where heat is absorbed to transform the solid CO2 into CO2 gas. This heat absorption at constant atmospheric pressure contributes to the enthalpy change of the system. However, not all the absorbed heat increases the internal energy, as some energy is used in doing work against the atmospheric pressure during the volume expansion of CO2 gas. This scenario gives rise to the phenomenon observed in the exercise, where the heat absorbed is more than the change in internal energy due to the energy expended in doing work.

Internal Energy Concept

The core of thermodynamics is the concept of internal energy, often symbolized as 'U'. It encompasses all the energy stored within a system, which can be altered through heating, cooling, or doing work. In our exercise, the internal energy of dry ice changes as it absorbs heat energy and sublimates into gas.

Internal energy is affected by both the temperature of the system and the state of the matter within it. As the temperature increases, the molecules move faster, and the internal energy rises. However, during a phase change like sublimation, even though the temperature of the substance remains constant, the internal energy increases because the particles' potential energy changes as they shift from a solid to a gaseous state. The first law of thermodynamics simplifies this complex interaction, as it relates the change in internal energy, the heat added to the system, and the work done by the system.