Problem 62 - Pearson Physics

Question

A fluid expands by \(0.42 \mathrm{~m}^{3}\) at a pressure of \(121 \mathrm{kPa}\). How much work is done by the fluid?

Step-by-Step Solution

Answer

The work done by the fluid is 50,820 J.

Step 1: Understand the Formula

Work done by or on a fluid during expansion or compression at constant pressure is calculated as the change in volume multiplied by the pressure. This is given by the formula: \[ W = P \Delta V \]where \(W\) is the work done, \(P\) is the pressure, and \(\Delta V\) is the change in volume.

Step 2: Convert Units

The pressure \(P\) given is in kilopascals (\(\mathrm{kPa}\)), and to maintain consistency with units in scientific calculations, we need to convert it to pascals (\(\mathrm{Pa}\)). Since \(1\, \mathrm{kPa} = 1000\, \mathrm{Pa}\), the pressure in pascals is:\[ P = 121 \times 1000 = 121,000\, \mathrm{Pa} \]

Step 3: Calculate the Work Done

Now, substitute the pressure in pascals and the change in volume into the formula for work:\[ W = 121,000\, \mathrm{Pa} \times 0.42\, \mathrm{m}^{3} \]Calculate to find the work done:\[ W = 50,820\, \mathrm{J} \]

Step 4: Express the Result

The work done by the fluid during the expansion is \(50,820\, \mathrm{J}\).

Key Concepts

Pressure-Volume Work

When a fluid expands, it performs work, which can be understood using the concept of pressure-volume work. Imagine a container with a piston fixed on top. As the fluid in the container heats up, it expands, pushing the piston upward. This movement is the essence of work done by the fluid. The formula used to calculate this work is:

• W = P \Delta V
• \(W\) represents the work done,
• \(P\) is the constant pressure the fluid is under,
• \(\Delta V\) is the change in volume.

The process of expansion under constant pressure is central to many applications like engines and respiratory systems. As the fluid expands and occupies more space, energy is transferred, allowing mechanical work to occur.

Unit Conversion

In the scientific world, ensuring that all units are consistent is crucial, which is where unit conversion comes in. For instance, pressure is often given in kilopascals (\(\mathrm{kPa}\)), while scientific calculations typically use pascals (\(\mathrm{Pa}\)). Converting units helps maintain consistency and avoid errors. Here’s how you convert \(\mathrm{kPa}\) to \(\mathrm{Pa}\):

• Know the conversion factor: \(1\, \mathrm{kPa} = 1000\, \mathrm{Pa}\).
• Multiply the given value in \(\mathrm{kPa}\) by 1000 to get \(\mathrm{Pa}\).

For example, if pressure is \(121\, \mathrm{kPa}\), converting to pascals gives us \(121,000\, \mathrm{Pa}\). This conversion ensures our calculations for work done are accurate and precise. Unit conversion is essential in engineering, physics, and many other fields where precision is key.

Fluid Dynamics

Fluid dynamics is the study of how fluids (liquids and gases) move. It's a branch of physics that deals with the motion of fluids and the forces involved. Understanding fluid dynamics is essential for calculating work done by fluids during expansion or contraction. Here are key principles in fluid dynamics:

• **Continuity Equation**: Ensures mass conservation, stating that the mass flow rate is constant in a pipe with variable diameters.
• **Bernoulli's Equation**: Describes energy conservation in a fluid flow, vital for understanding pressure changes in a flowing fluid.
• **Laminar vs. Turbulent Flow**: Determines how smoothly or chaotically a fluid moves through a space.

Fluid dynamics is applied in various fields including aviation, weather prediction, and hydraulic systems. By understanding these principles, one can better grasp how fluids perform work when engaging in processes like expansion or compression.