91Ó°ÊÓ

Understanding an isothermal process is crucial for solving problems like this one. An isothermal process means the temperature remains constant throughout the process. So, in our example, no significant heat transfer occurs between the tank and surroundings.
An isothermal process can be described by Boyle's Law, which states that for a given mass of an ideal gas, the product of pressure (P) and volume (V) is constant when the temperature is kept constant: \[ PV = \text{constant} \]
Since temperature doesn't change in an isothermal process, the final temperature in the tank will be the same as the surroundings, which is 25°C or 298.15 K in this problem.

The molar mass of air is the average mass of the air's molecules. Air is primarily composed of nitrogen (N_2), oxygen (O_2), argon (Ar), and trace amounts of other gases.
The molar mass of air is calculated using the weighted average based on the abundance of these gases. For simplification in our calculations, the molar mass of air is often approximated as 29 g/mol.
This molar mass is used to convert the number of moles of air to the mass of air. For example, if we have 0.404 moles of air (as calculated using the ideal gas law), the mass can be found by multiplying the number of moles by the molar mass:\[ \text{Mass} = n \times \text{Molar Mass} \] This approach helps us find that 0.404 moles multiplies by 29 g/mol results in approximately 11.72 grams of air.

A rigid tank problem involves finding out how gases behave within a fixed volume container when subjected to changes in pressure or temperature. In our problem, the tank’s volume is constant at 10 liters (0.01 m³).
Since the tank doesn't change in size (hence 'rigid'), any change in the quantity of gas or its state parameters (like pressure or temperature) can be directly assessed using the ideal gas law. Given, the process is isothermal (constant temperature) and the pressure inside the tank eventually balances with the surroundings at 1 bar, we can simplify our calculations significantly.
The ideal gas law formula, \( PV = nRT \) allows us to solve for the unknowns (such as the number of moles of air that leaks in) since the volume (V) and the pressure (P) are known, and the gas constant (R) and temperature (T) are standard values (R = 8.314 J/mol·K and T = 298.15 K).
This results in the final mole count for air which we can then convert to mass using the air's molar mass. All these steps must consider the rigid nature of the tank, ensuring no volume change impacts our calculations.