Based on the theory of the molecular model of an ideal gas (for the kinetic theory of gases) show that the total pressure exerted by the gas on the walls of the container can written as \[ P=\frac{2}{3}\left(\frac{N}{V}\right)\left(\frac{1}{2} m \bar{v}^{2}\right) \]

In the kinetic theory of gases, we start by understanding how gas molecules collide with the walls of a container. Each molecule has an average kinetic energy that can be expressed in terms of its velocity. By considering the motion of gas molecules and counting their collisions with the walls, we derive the expression for pressure, \( P \). The pressure is proportional to the number density (\( N/V \)) and the average kinetic energy per molecule, which is given as \( \frac{1}{2} m \bar{v}^{2} \)—it all comes together to show that gas pressure emerges from these molecular interactions! This equation demonstrates that pressure isn't just a simple result of mass but rather the dynamic motion of countless molecules bouncing around. In practical applications, this understanding is pivotal in designing engines, refrigeration systems, and even air conditioning units. Engineers leverage this principle to optimize the performance of various systems, using the kinetic vision of gases to influence designs that require precise control over pressure and volume interactions for maximum efficiency!