91Ó°ÊÓ

The Ideal Gas Law is a fundamental equation in chemistry and physics. It relates the pressure, volume, temperature, and number of moles of an ideal gas. Written as \[[ PV = nRT ]\], the law allows us to predict one state variable if the others are known. Here:
- \( P \) is the pressure in the tank
- \( V \) is the volume of the tank
- \( n \) is the number of moles of gas
- \( R \) is the ideal gas constant
- \( T \) is the temperature in Kelvin
For our problem, the final pressure, volume, and temperature are given. We can rearrange the law to solve for the number of moles, \( n \): \[ n = \frac{PV}{RT} \]. By substituting the known values, we calculate the number of moles of gas added to the tank through the pinhole.

Thermodynamics is the scientific study of energy, heat, and work. It includes various laws that describe how energy moves and changes in a system. In this problem, we are dealing with a fundamental part of thermodynamics: the behavior of gases under different conditions.
Key concepts:
- Energy can be transferred as heat or work
- Energy is conserved< br>The Ideal Gas Law is one of the tools used in thermodynamics to model the behavior of gases. Understanding how gases expand, compress, and transfer energy is crucial in many scientific and engineering applications. For this exercise, knowing thermodynamics helps us determine how much work was done by the gas when it entered the tank.

The First Law of Thermodynamics is also known as the Law of Energy Conservation. It states that energy cannot be created or destroyed, only transferred or converted from one form to another. Written as \[ \Delta U = Q - W ]\], where:
- \( \Delta U \) is the change in internal energy
- \( Q \) is the heat added to the system
- \( W \) is the work done by the system
This law is particularly useful in analyzing closed systems where mass does not enter or leave.
In this problem, the air enters the tank, and because the process is isothermal, \( \Delta U = 0 \). Therefore, we have \( Q = W \), implying that the heat added to the tank is entirely converted into work done by the gas as it enters. Given the process conditions (isothermal), calculation steps are simplified since internal energy change is zero.

An isothermal process is when a system's temperature remains constant during gas expansion or compression. For ideal gases, this means that any heat added to the system will convert entirely into doing work.
Key features:
- Temperature (\( T \)) is constant
- Internal energy (\( \Delta U \)) does not change
- Ideal gases follow Part of thermodynamic transformations
In our problem, the temperature inside the tank remains at 21°C (294.15 K). Because it's isothermal, we do not have to worry about changes in internal energy. The heat added to the gas as it enters the tank is used entirely to do work. Thus, applying the First Law of Thermodynamics simplifies our calculations concerning the heat transfer.