91Ó°ÊÓ

The Ideal Gas Law is a fundamental concept in thermodynamics. It's expressed as \( PV = nRT \), where:

• \( P \) represents pressure
• \( V \) is volume
• \( n \) is the number of moles
• \( R \) is the universal gas constant
• \( T \) is the temperature in Kelvin

In this exercise, we are dealing with a situation where the pressure remains constant, allowing us to use the concept that volume (V) is directly proportional to temperature (T) when rearranging the equation. This relationship helps to determine how much the volume changes as the temperature of an ideal gas is increased or decreased at a constant pressure. By understanding this relationship, we can predict the behavior of the gas under different conditions. This law is crucial for solving problems involving gases because it connects macroscopic properties like pressure, temperature, and volume, making it possible to solve for one when the others are known.

Gas expansion is an important process in thermodynamics, particularly when dealing with ideal gases. It involves the increase in volume of a gas as its temperature rises or pressure decreases, or both. In our particular case, the gas expands because the temperature is increased while keeping pressure constant. For a constant pressure process, the gas expands more as the temperature increases. During this expansion, we are interested in how the final volume is affected by the change in temperature. With the assumption of ideal gas behavior, we can use the equation \( V_2 / V_1 = T_2 / T_1 \) to find the final volume \( V_2 \). This expression shows us that the final and initial volumes of the gas are directly proportional to their respective temperatures when the pressure is constant. Understanding how gases expand and contract under varying conditions is foundational in predicting and manipulating gas behavior in both scientific and industrial applications.

Temperature conversion is a straightforward yet crucial step when working with thermodynamics and the ideal gas law. The rule of thumb is to always convert temperatures from Celsius to Kelvin before doing any calculations that involve the physical properties of gases. The Kelvin scale is preferred because it is an absolute temperature scale, which means it starts at absolute zero, where all molecular motion theoretically stops. This makes it ideal for calculations involving temperature relations.The conversion is simple:

• To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.

In our exercise, we start with initial and final temperatures in Celsius: \(27^{\circ} \text{C}\) and \(227^{\circ} \text{C}\). Converting them gives us 300.15 K and 500.15 K respectively. Always remember to check units—using the wrong scale can lead to incorrect conclusions in your calculations.

Work calculation in thermodynamics is an essential aspect, often related to energy transformations. In our exercise, we calculate the work done by an expanding gas at constant pressure using the formula \( W = P \Delta V \), where:

• \( W \) is the work done by the gas.
• \( P \) is the pressure, which must be converted to appropriate units (Pascals in our case).
• \( \Delta V \) is the change in volume, found by subtracting the initial volume from the final volume.

Here the gas expands from 3.0 L to 5.0 L, so \( \Delta V \) is 2.0 L which we convert to cubic meters for calculation (0.002 m³). We then multiply by the pressure expressed in Pascals—2 atm equals 202650 Pa—for accurate work computation. Finally, the calculation yields a work done of 405.3 J. Understanding these calculations helps us grasp how mechanical work and thermal energy exchange are related in physical systems.