Problem 64 - Pearson Physics

Question

A gas expands at a constant pressure of \(190 \mathrm{kPa}\) and does 82 J of work. What was the change in volume?

Step-by-Step Solution

Answer

The change in volume is approximately 0.000432 m³ or 432 cm³.

Step 1: Understanding the problem

We need to find the change in volume (\( \Delta V \)) of a gas that expands under constant pressure. Given the constant pressure (\( P = 190 \mathrm{kPa} \)) and the work done (\( W = 82 \mathrm{J} \)), we will use the formula for work done by a gas during expansion.

Step 2: Using the Work Formula

The work done by the gas during expansion at constant pressure is given by the formula \( W = P \Delta V \). We substitute the values we have into this formula, keeping in mind that \( P \) should be in Pascals. So, \( 190 \mathrm{kPa} \) = \( 190,000 \mathrm{Pa} \).

Step 3: Solving for the Change in Volume

Rearrange the formula \( W = P \Delta V \) to solve for \( \Delta V \), which gives \( \Delta V = \frac{W}{P} \). Substitute the values \( W = 82 \) J and \( P = 190,000 \) Pa into the equation to find \( \Delta V \).

Step 4: Calculation

Calculate \( \Delta V = \frac{82}{190,000} = 0.000431579 \, \mathrm{m^3} \), or \( 431.579 \, \mathrm{cm^3} \) using the conversion 1 cubic meter = 1,000,000 cubic centimeters.

Key Concepts

Gas Expansion

When we talk about gas expansion, we are referring to the process where a gas increases its volume. This can happen in different ways, such as when the temperature of the gas rises or when the gas is allowed more space, such as moving a piston in an engine.
During expansion, the gas molecules move more freely and occupy a larger volume. - Gas expansion can be isothermal (constant temperature), adiabatic (no heat exchange), or isobaric (constant pressure). - In this exercise, the gas undergoes an isobaric expansion, meaning the pressure stays the same throughout the process. Understanding gas expansion is key in thermodynamics, as it helps explain how energy is transferred in systems and is also crucial for designing engines and refrigerators.

Work Done by Gas

The concept of work in physics refers to the transfer of energy when a force causes an object to move. In thermodynamics, work done by gas is the energy transferred as the gas expands or contracts. Using the formula for work done by gas: \( W = P \Delta V \)- \( W \) is the work done by the gas.- \( P \) is the constant pressure.- \( \Delta V \) is the change in volume of the gas.In this scenario, the gas does 82 J of work while expanding. The work done is an indication of how much energy the gas molecules exert in pushing against the external pressure. This concept is crucial for understanding energy conversion processes in heat engines.

Constant Pressure Expansion

In constant pressure expansion, or isobaric expansion, the pressure remains steady while the volume of gas changes. This is a common scenario in engines and air conditioning systems where the gas expands against a constant external pressure.
The relationship between work, pressure, and volume change is linear in this case, thanks to the formula: \( W = P \Delta V \).Some key points to consider:- Constant pressure makes it simple to calculate work done by simply multiplying the pressure by the volume change.- Maintaining constant pressure often requires adding or removing heat from the system to offset the energy transferred as work. This type of expansion illustrates a fundamental principle of thermodynamics, where energy transfers due to volume changes under fixed conditions.

Volume Change Calculation

Calculating the change in volume due to gas expansion involves using the formula that connects work, pressure, and volume change: \[\Delta V = \frac{W}{P}\]By substituting known values into the equation, you can solve for the unknown variable. In this exercise:- Work \( W = 82 \) J.- Pressure \( P = 190,000 \) Pa (as 1 kPa = 1000 Pa).Then, simply plug in these values:\[\Delta V = \frac{82}{190,000} \]After calculating, you find that the volume change is approximately 0.000431579 cubic meters.
This can also be expressed as 431.579 cubic centimeters, applying the conversion (1 cubic meter = 1,000,000 cubic centimeters).Volume change calculations are vital in numerous applications, including the design and analysis of engines and industrial processes where gas is used or produced.