91Ó°ÊÓ

The Ideal Gas Law is a cornerstone of thermodynamics, providing a relationship between the pressure, volume, and temperature of an ideal gas. It's often expressed as \( PV = nRT \), where:

• \( P \) is the pressure of the gas
• \( V \) is its volume
• \( n \) is the number of moles
• \( R \) is the ideal gas constant
• \( T \) is the absolute temperature

When dealing with processes where the initial and final conditions of a gas are known, this equation helps us calculate unknown parameters. For instance, when a gas is heated in a tank, the pressure can be determined by using the initial and final states of temperature and the volume of the gas remains unchanged in a rigid tank. This relationship was pivotal in calculating the pressures at different states in the given exercise.

An isothermal process is a type of thermodynamic process where the temperature remains constant. This means that any heat added to the system is used to do work. In the case of our exercise, when one side of the tank acts as a piston, allowing the air to expand, the process that occurs is isothermal.

This is particularly interesting because the ideal gas law simplifies during an isothermal process: since \( T \) is constant, the product \( PV \) remains constant. For example, if the volume of the gas doubles in an isothermal expansion, the pressure must halve, given that \( P_2 V_2 = P_3 V_3 \). This process was used to determine how pressure changes when the volume of the tank expands.

Heat transfer is an essential concept in thermodynamics, indicating the exchange of thermal energy between systems. In our exercise, this energy transfer happens twice.

First, when the air in the rigid tank is heated, all the energy goes into increasing the internal energy since the volume does not change (implying no work is done until expansion begins). This heat transfer is calculated using specific heats at constant volume, \( C_v \), and the change in temperature.

• For rigid tanks, it is calculated as: \[ Q_{1-2} = nC_v(T_2 - T_1) \]

Then during the isothermal expansion, the heat added to the system is completely converted into work, maintaining the temperature constant.

Work done in a thermodynamic process translates to energy transferred. In the context of our problem, work is done during the isothermal expansion of air.

For an isothermal process, work done is calculated using the formula:\[ W = nRT \, \ln \left( \frac{V_3}{V_2} \right) \]This equation tells us how much energy is spent in volume expansion without a temperature change. The natural log ensures that even if distinct volumes are considered, continuity in pressure and energy distribution is maintained. This concept offers insight into how energy is neither lost nor gained in an isothermal state, maintaining the system equilibrium as long as the external conditions change uniformly.

A rigid tank is a constant volume container often used in thermodynamic problems. Its main characteristic is that the volume doesn't change, even when pressure and temperature vary.

In our exercise, the rigid tank dictates the first part of the process where the air is heated with the volume held steady. This means no work can be done because work requires a change in volume (\( W = P \Delta V \)). Hence, heating directly influences the internal energy, resulting in a change in pressure with temperature as described by the Ideal Gas Law. Understanding how a rigid tank affects a thermodynamic system helps solve problems related to variable pressure and temperature states without volume fluctuation.