For the isothermal expansion of a gas into a vacuum, ΔE = 0, q = 0, and w = 0. (b) Explain why no work is done by the system during this process.

Verified step by step guidance

1. insert step 1> Understand the concept of isothermal expansion into a vacuum, which is also known as free expansion.
2. insert step 2> Recognize that in a vacuum, there is no external pressure opposing the expansion of the gas.
3. insert step 3> Recall that work done by a gas during expansion is given by the formula: w = -P_{ext} \Delta V, where P_{ext} is the external pressure.
4. insert step 4> Since the gas is expanding into a vacuum, P_{ext} = 0, meaning there is no opposing force.
5. insert step 5> Conclude that with P_{ext} = 0, the work done, w, is also 0, as w = -0 \times \Delta V = 0.

Key Concepts

Isothermal Process

An isothermal process occurs at a constant temperature, meaning that the internal energy of an ideal gas remains unchanged during the expansion or compression. In this scenario, since the temperature is constant, any heat added to the system is used to do work, but in the case of expansion into a vacuum, there is no heat exchange with the surroundings.

Work in Thermodynamics

In thermodynamics, work is defined as the energy transfer that occurs when a force is applied over a distance. For a gas expanding against an external pressure, work is done by the system. However, in an expansion into a vacuum, there is no opposing pressure, meaning the gas does not push against anything, resulting in zero work done.

First Law of Thermodynamics

The First Law of Thermodynamics states that energy cannot be created or destroyed, only transformed. In the case of the isothermal expansion into a vacuum, the change in internal energy (ΔE) is zero, indicating that the energy remains constant. Since no work is done and no heat is exchanged, the system's energy balance confirms that ΔE = q + w = 0.