Explain in terms of the kinetic theory how raising the temperature of a confined gas makes its pressure increase.

Imagine a crowd of people in a room moving at different speeds. Some are wandering slowly, while others might be sprinting. In the kinetic theory of gases, temperature is like the overall energy buzz of that crowd. It describes how quickly gas particles are moving on average. When we talk about temperature in relation to gases, we're really discussing the average kinetic energy of all the particles.

As the temperature increases, this average energy goes up, much like how the room's energy would increase if more people started running. In more scientific terms, the relationship between temperature and kinetic energy is given by the equation: \( KE_{avg} = \frac{3}{2}k_BT \), where \(KE_{avg}\) is the average kinetic energy of the gas particles, \(k_B\) is Boltzmann's constant, and \(T\) is the temperature in kelvins. Simply put, a hotter gas has particles that, on average, move faster because they possess more kinetic energy.

Pressure can be a bit like the sound volume in a room full of those moving people. The louder the room, the more energy there is buzzing around. For a gas, the pressure is essentially the collective 'punch' the gas particles deliver when they hit the walls of their container. When the average kinetic energy of the particles increases with temperature, these particles start hitting the walls more often and with greater force.

The pressure of a gas is calculated using the equation: \( P = \frac{F}{A} \), where \(P\) is the pressure, \(F\) is the force exerted by the particles on a surface of the container, and \(A\) is the area of that surface. As the temperature rises, the average speed and therefore the force of impact on the walls increases, leading to an increased pressure. If the volume of the gas doesn't change because it's confined, this effect is even more pronounced, as we'll see in the next section.

As the energized gas particles zip around, they're bound to bump into each other and the container walls, much like people moving in a busy space. These particle collisions are the heart of the kinetic theory of gases. Not only do they account for the gas pressure as mentioned earlier, but they also result in the spread of energy among particles, a process known as thermal equilibrium.

Impact of Collisions

When the temperature of the gas is raised, these collisions with the container walls become more frequent and vigorous. These constant 'mini-explosions' against the container's walls are what we measure as pressure. It's like the difference between walking into a wall as opposed to running into it; the faster the motion (or higher the temperature), the more intense the collision.

Collisions and Volume

In a confined space, there's less 'wiggle room' for the particles, leading to even more collisions. This is especially important to consider when working with gases in a closed system where volume is constant. Here, any increase in temperature doesn't have the outlet of expansion, so pressure builds up due to the heightened collision rate.