91Ó°ÊÓ

Mass flow rate is a critical concept in fluid mechanics. It measures the amount of mass passing through a given point per unit time.
In this exercise, the mass flow rate of air leaking into the tank is given as \(\dot{m} = 0.002 \text{ kg/s}\).
This constant flow rate helps us determine how much air (by mass) enters the tank over a specified period. To calculate the total mass entering the tank after 30 seconds, simply multiply the mass flow rate by the time: \(\ m = \dot{m} \ t = 0.002 \ kg/s \ \times 30 s = 0.06 \ kg \).
It's important to understand that controlling and calculating mass flow rates is pivotal in many engineering applications, including aerodynamics, chemical reactions, and HVAC systems.

Calculating the pressure in the tank after a certain time involves using the ideal gas law, which relates the pressure, volume, and temperature of a gas to the amount of gas present in moles.
The ideal gas law is stated as: \[ PV = nRT \]
where, \ P = \text{pressure (Pa)}
V = \text{volume (cubic meters)}
n = \text{number of moles}
R = \text{ideal gas constant (8.314 J/mol·K)}
T = \text{temperature (Kelvin)}
However, we already know the mass and wish to calculate the pressure.
Thus, we use the modified form of the ideal gas law: \[ P = \frac{mRT}{MV} \]
where \ M is the molar mass of air.
Given: \ m = 0.06 \ kg, V = 0.15 \ \text{m}^3, T = 293.15 \ \text{K}, M = 0.029 \ \text{kg/mol} \ and R = 8.314 \ \text{J/mol·K},
we substitute these values into our formula to find the pressure: \[ P = \frac{0.06 \ kg \ \times 8.314 \ J/mol·K \ \times 293.15 \ K }{0.029 \ kg/mol \ \times 0.15 \ \text{m}^3} ≈ 33.77 \ \text{kPa} \]
Calculations like these demonstrate the practical applications of fundamental physics principles.

Thermodynamic properties are characteristics of systems that dictate the state and behavior of a material under various conditions.
In this exercise, relevant properties include temperature, pressure, volume, and mass.
Understanding these properties and the relationships between them is essential.
For example, temperature is a measure of the average kinetic energy of the molecules in a substance. It's measured in Kelvin when using the ideal gas law.
Pressure is the force exerted by the gas per unit area, important in determining the state of the gas within the tank.
Volume is the amount of space the gas occupies, which in this problem is given as 0.15 m³.
Mass is the amount of matter in the gas, calculated using the mass flow rate and time.
These properties are interconnected through the ideal gas law, providing a clear relationship geared towards predicting the behavior of gases under various conditions.
Developing a profound understanding of these thermodynamic properties ensures comprehensive insight into gas behaviors in practical and theoretical situations.