Question

What is the work when a gas contracts from \(3.45 \mathrm{~L}\) to \(0.97 \mathrm{~L}\) under an external pressure of \(0.985 \mathrm{~atm} ?\)

Short answer

The work done on the gas is 2.4428 L·atm.

Step-by-step solution

Understanding the Work Formula for Gases

When a gas contracts under constant external pressure, the work done by or on the gas can be calculated using the formula: \[ W = -P_{ext} \times \Delta V \] where \( W \) is the work done, \( P_{ext} \) is the external pressure, and \( \Delta V \) is the change in volume.

Calculate the Change in Volume

The change in volume \( \Delta V \) can be calculated as the final volume minus the initial volume. Here, the initial volume \( V_i = 3.45 \mathrm{~L} \) and the final volume \( V_f = 0.97 \mathrm{~L} \). Thus, \[ \Delta V = V_f - V_i = 0.97 - 3.45 = -2.48 \mathrm{~L} \] The negative sign indicates a contraction.

Substitute Values into the Work Formula

Now that we have \( \Delta V = -2.48 \mathrm{~L} \) and \( P_{ext} = 0.985 \mathrm{~atm} \), we substitute these values into the work formula: \[ W = -0.985 \times (-2.48) \]

Calculate the Work

Perform the multiplication in the formula: \[ W = 0.985 \times 2.48 = 2.4428 \mathrm{~L} \cdot \mathrm{atm} \] The work done is positive, indicating work done on the gas (as it contracts).

Key concepts

Gas Contraction

Gas contraction occurs when a gas reduces its volume. This typically happens because the molecules of the gas are getting closer together, occupying a smaller space. Such contraction might be a result of cooling, which reduces the energy of the gas molecules, allowing them to move less and settle closer to each other. Another cause can be an increase in external pressure, which pushes the gas molecules closer together. When gases contract, they often do work on the surroundings by releasing energy. Understanding this can help predict how gases will behave under different conditions, such as in engines and weather systems.

External Pressure

External pressure is the force exerted by surrounding substances or environments upon the gas. This pressure can come from various sources, such as the weight of air in the atmosphere or a piston within a cylinder. The role of external pressure is crucial in determining how gases will behave in confined spaces.
- It impacts the physical state of the gas, potentially causing it to contract or expand. - In the context of thermodynamic work, external pressure is a key factor in calculating how much work is performed. In the original exercise, understanding the concept of external pressure helps us identify how much force is acting on the gas, which helps calculate the work done during the gas contraction.

Change in Volume

The change in volume is represented as the difference between the final volume and the initial volume of the gas. This change is a crucial component in determining the work done when a gas contracts under a constant external pressure. To calculate the change in volume, you simply subtract the final volume from the initial volume:

- Initial Volume ( V_i ): The starting volume of the gas before contraction. - Final Volume ( V_f ): The volume of the gas after the contraction has occurred.

Using the formula, \( \Delta V = V_f - V_i \), we can easily find the change in volume. In many scenarios, a negative change in volume indicates contraction, requiring careful attention in calculations.

Work Formula for Gases

The work formula for gases is central to understanding how work is calculated when a gas contracts or expands under a constant external pressure. The formula is:\[ W = -P_{ext} \times \Delta V \]Where:

- \( W \): Represents the work done. - \( P_{ext} \): External pressure acting on the gas. - \( \Delta V \): Change in volume of the gas.

This formula arises from the principles of thermodynamics. The negative sign indicates that work done by the system is negative, while work done on the system is positive. Thus, if the volume change is negative (as in contraction), the resulting work value will be positive, signifying work done on the gas. Applying this to problems helps accurately predict energy changes when a gas is subjected to external forces.