Problem 57 - General Chemistry: Principles and Modern Applications

Question

The internal energy of a fixed quantity of an ideal gas depends only on its temperature. A sample of an ideal gas is allowed to expand at a constant temperature (isothermal expansion). (a) Does the gas do work? (b) Does the gas exchange heat with its surroundings? (c) What happens to the temperature of the gas? (d) What is \(\Delta U\) for the gas?

Step-by-Step Solution

Answer

(a) The gas does work. (b) The gas does exchange heat with its surroundings. (c) The temperature of the gas remains constant. (d) \(\Delta U = 0\)

Step 1: Does the gas do work?

In an isothermal expansion, the ideal gas does do work. This is because when the gas expands, it is pushing outward against the pressure of its surroundings, causing it to do work. This is in accordance with the gas law \(W = P \Delta V\), where \(P\) is pressure and \(\Delta V\) is the change in volume. If there's an expansion, there's a positive \(\Delta V\).

Step 2: Does the gas exchange heat with its surroundings?

Yes, in an isothermal expansion, the ideal gas does exchange heat with its surroundings. Because the internal energy of the gas does not change due to the constant temperature, the work done by the gas has to be balanced by an equal amount of heat absorbed from the surroundings. This is in accordance with the first law of thermodynamics, \(\Delta U = Q - W\), if \(\Delta U=0\), then \(Q = W\).

Step 3: What happens to the temperature of the gas?

The temperature of the gas remains constant during an isothermal expansion. This is inherent in the definition of an 'isothermal' process, which states that the process occurs at a constant temperature. Therefore, no matter how much the gas expands or how much work it does, the temperature of the gas remains the same.

Step 4: What is \(\Delta U\) for the gas?

\(\Delta U\), the change in internal energy of the gas, is zero. As mentioned in the question, the internal energy of an ideal gas depends only on its temperature. In an isothermal process, since the temperature of the gas remains constant, there is no change in internal energy: \(\Delta U = 0\).

Key Concepts

Ideal Gas

An ideal gas is a simplified model that helps us understand the behavior of gases. The concept assumes that gas molecules have no volume and do not attract or repel each other. This simplification makes calculations and predictions easier, especially when studying processes like expansion and compression.
In reality, no gas is "ideal," but many common gases like oxygen and nitrogen behave similarly to an ideal gas under typical conditions of pressure and temperature. The idea of an ideal gas is grounded in the equation known as the Ideal Gas Law:

• \( PV = nRT \)
• Here, \(P\) stands for pressure, \(V\) is volume, \(n\) represents the number of moles of gas, \(R\) is the ideal gas constant, and \(T\) is the temperature in Kelvin.

Understanding the Ideal Gas Law is fundamental to explaining how gases react under various conditions without focusing on the intricate details of molecular interactions, making it a crucial tool for studying thermodynamics.

First Law of Thermodynamics

The First Law of Thermodynamics is a foundational concept that describes the conservation of energy. It essentially states that energy cannot be created or destroyed, only transformed from one form to another. This idea is expressed through the equation:

• \( \Delta U = Q - W \)
• \( \Delta U \) represents the change in internal energy of the system.
• \( Q \) is the heat added to the system.
• \( W \) is the work done by the system on its surroundings.

In the case of isothermal expansion of an ideal gas, where temperature stays constant, the internal energy does not change. This means \( \Delta U = 0 \). Therefore, according to the First Law of Thermodynamics, any work done by the gas must be matched by an equivalent amount of heat absorbed. In simple terms, whatever energy the gas loses or uses in doing work, it gains right back as heat from its surroundings.
This elegant law helps us understand the exchange of energy within a system, ensuring balance in every thermal event.

Internal Energy

Internal energy is a term that describes the total energy contained within a system due to the motion and arrangement of its molecules. For an ideal gas, this internal energy is solely dependent on its temperature. This means that if the temperature of the gas doesn't change, its internal energy remains constant.
During an isothermal expansion, since the temperature of the gas does not change, the internal energy also does not change. As per the exercise you’re working on, this is expressed as:

• \( \Delta U = 0 \)

This concept is particularly useful because it simplifies calculations in thermal physics. Knowing that the internal energy in an ideal isothermal process doesn’t change allows us to predict the behavior of gases under stable temperature conditions with ease. It channels our focus on other elements like heat transfer and work done, which actively participate in changing the system's state.
Internal energy, in essence, acts as a clear indicator of thermal conditions, guiding you through understanding the dynamic nature of gas behavior.