Question

A \(1.00 \mathrm{g}\) sample of \(\mathrm{Ne}(\mathrm{g})\) at 1 atm pressure and \(27^{\circ} \mathrm{C}\) is allowed to expand into an evacuated vessel of \(2.50 \mathrm{L}\) volume. Does the gas do work? Explain.

Short answer

Yes, the gas does work during the process. You can deduce this by identifying the initial and final states of the system, determining the change in volume, and then calculating the work done by the system. The work done as calculated will be negative, indicating that the gas did work on the surroundings during its expansion.

Step-by-step solution

Identify the initial state

Let's identify the initial state. Here, the \(1.00 \mathrm{g}\) sample of \(\mathrm{Ne}(\mathrm{g})\) at 1 atm pressure and\(27^{\circ} \mathrm{C}\) forms our initial state.

Identify the final state

Now let's identify the final state. When the gas is allowed to expand into an evacuated vessel of \(2.50 L\), this forms our final state.

Determine the change in volume

Since the initial state is in a container that is filled to a certain volume, and the final state is in an evacuated vessel of a known larger volume, the volume change \(\Delta V\) will be the final volume of the vessel.

Calculate the work done

Work done \(W\) by the gas can be calculated using the formula \( W = -P\Delta V\), where \(P\) is the pressure the gas initially had and \(\Delta V\) is the change in volume. Recall that the work done by gas on its surroundings in an isothermal (constant temperature) expansion is negative, as it uses its internal energy to push against the external pressure.

Conclusion

If the calculated work done is more than zero, the gas did work while expanding into the evacuated volume.

Key concepts

Gas Expansion

Gas expansion occurs when gas molecules spread out into a larger volume. Imagine a balloon inflating; that's a visual representation. In our exercise, the neon gas goes from a compact space into a bigger vessel. The initial state was at 1 atm pressure and 27°C, while the final state was with a volume of 2.5 L in an evacuated container.
The key here is how gases move when given space. Gas molecules are always in motion. They naturally spread out to fill the available volume.

• Initial volume: small and compact.
• Final volume: larger, as the gas fills the new container.

Understanding gas behavior is crucial. This spreading out isn’t random; it follows specific physical principles.Think of gas expansion as nature's way of balancing things out.

Isothermal Processes

During an isothermal process, the temperature of the gas remains constant. This is essential when analyzing gas expansion. The exercise involves neon gas expanding at a constant 27°C. This constant temperature ensures that we focus on volume change, not temperature change.
Why is temperature constant? The system may exchange heat with its surroundings to maintain it. This balance keeps the temperature steady.
In real-life scenarios, an isothermal process means:

• No change in temperature during the reaction.
• Heat may flow in or out, but doesn't change the thermal energy within the gas.

The magic of isothermal processes lies in their simplicity. Despite volume and pressure changes, the temperature's constancy simplifies calculations and predictions about the gas behavior.

Work Done by a Gas

Work done by a gas is a core concept in thermodynamics. It is a measure of energy transfer as the gas expands or contracts. Using the formula \( W = -P\Delta V \), we compute the work done by the gas.

Here's a breakdown of the formula:

• \(W\) is the work done.
• \(P\) is the pressure.
• \(\Delta V\) is the change in volume.

In an expansion, the work is often negative. This is because the gas uses energy to push against the external environment. The sign reminds us about the direction of energy transfer.
Applying this to our scenario, as the neon gas expands into an evacuated vessel, it essentially does not do work on the surroundings. There's no external pressure acting against it, so \(P\) is effectively zero. Even if there's a volume change, the absence of pressure means no work is done, reinforcing the principle that gases require resistance to perform work.