Problem 11 - Chemistry

Question

On what law is the first law of thermodynamics based? Explain the sign conventions in the equation $$ \Delta U=q+w $$

Step-by-Step Solution

Answer

The first law of thermodynamics is based on energy conservation. Positive \( q \) means heat is absorbed, negative \( q \) means released; positive \( w \) means work is done on the system, negative \( w \) means the system does work.

Step 1: Understanding the First Law of Thermodynamics

The first law of thermodynamics is based on the principle of the conservation of energy. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. In the context of thermodynamics, this means the change in internal energy of a system is equal to the heat added to the system plus the work done on the system.

Step 2: Describing the Equation

The equation \( \Delta U = q + w \) represents the first law of thermodynamics, where \( \Delta U \) is the change in internal energy of a system, \( q \) is the heat exchanged with the surroundings, and \( w \) is the work done on or by the system.

Step 3: Explaining the Sign Convention for Heat (q)

According to thermodynamic conventions, \( q \) is positive when heat is absorbed by the system from the surroundings (endothermic process). Conversely, \( q \) is negative when heat is released by the system to the surroundings (exothermic process). This convention helps distinguish between heat flow directions.

Step 4: Explaining the Sign Convention for Work (w)

For work, \( w \) is positive when work is done on the system by the surroundings, meaning energy is being added to the system. Conversely, \( w \) is negative when work is done by the system on the surroundings, indicating energy is being lost by the system to its surroundings.

Key Concepts

Conservation of Energy

The first law of thermodynamics is closely related to the principle of the conservation of energy. This fundamental principle asserts that energy in a closed system is neither created nor destroyed. It can only change from one form to another, maintaining the total amount of energy constant. For example, the chemical energy in a battery can transform into electrical energy, but the total energy remains unchanged.
Understanding this principle is crucial when studying thermodynamics because it establishes the groundwork for how systems interact with their environment. It emphasizes that any energy gained or lost by a system must have come from or gone to its surroundings. In the context of a thermodynamic system, every gain or loss, whether as heat or work, corresponds to changes elsewhere in the energy network.
The conservation of energy is a reassuring rule—it tells us that no energy mysteriously disappears; it only changes form.

Internal Energy

Internal energy is a fundamental concept in thermodynamics referring to the total energy contained within a system. This encompasses all microscopic energies such as translational, rotational, vibrational, and any potential energies among molecules.
When a system experiences a change due to heat exchange or work, its internal energy also changes accordingly. The change in internal energy, denoted as \( \Delta U \), reflects these modifications. A positive \( \Delta U \) indicates that the system's internal energy has increased, typically due to absorbing heat or work done on it. On the other hand, a negative \( \Delta U \) implies the system has lost energy.
Grasping the concept of internal energy helps us understand how systems respond to interactions like heating or compression.

Heat Exchange

Heat exchange is the process of energy transfer between a system and its surroundings due to a temperature difference. It is represented by \( q \) in the first law of thermodynamics equation \( \Delta U = q + w \).
There are specific conventions for understanding heat exchange:

• When heat flows into a system, it is labeled as positive \( q \). This usually happens when the system absorbs heat, such as when heating it up in an oven.
• Conversely, when a system releases heat to its surroundings, this is depicted by a negative \( q \). An example can be seen when cooling hot coffee.

These conventions enable clear communication and analysis of how energy moves in thermal processes, ensuring we understand when energy is being absorbed or lost in a system.

Work Done

In thermodynamics, work done, symbolized by \( w \), represents another way energy can be transferred between a system and its environment. It can occur through mechanisms like moving a piston in an engine or compressing a gas.
The first law's equation \( \Delta U = q + w \) assigns a sign to work done:

• When work is done on a system (e.g., compressing a gas by a piston), energy increases, and \( w \) is considered positive.
• If a system performs work on its surroundings (e.g., expanding gas moving a piston), \( w \) becomes negative, indicating a loss of energy from the system.

This understanding helps in analyzing energy exchanges in various thermodynamic processes, allowing predictions of how a system's internal energy will change based on its interactions.