Where does the force from increased collisions come from?

Say we have an ideal gas in a container and we raise its temperature through conduction. The Ideal Gas Law states that when temperature increases, pressure increases. Therefore, since pressure is the number of collisions divided by area and the area of the container is constant, we know that the number of collisions have increased.

I assumed that the increase in # of collisions is because of increased acceleration of each particle. And, with the mass of the gas staying constant, there should be a force causing particles to accelerate, according to Newton's 2nd Law.

However, where does that force come from? Thanks so much!

• The ideal gas law does not give a mechanism for this transfer process, but just describes the end result. Since temperature is intimately related to kinetic energy for ideal gases, the law just tells you how increasing temperature increases pressure, which you already noted means more/harder collisions of the particles with their container walls. Sep 18, 2023 at 0:18
• Hi! Thanks so much for your comment! I tried to derive the mechanism from the ideal gas law, but as you said, it doesn't show the process for the behavior of ideal gases. Where could I find the mechanism, or the force, that causes the increased pressure and acceleration? Thanks! Sep 18, 2023 at 0:24
• Since this is a thermodynamics process, there are always two potential sources: heat and mechanical work. Depending on the details of the process used, heat, work, or some combination thereof may be responsible for the transfer of energy from the surroundings to the gas molecules. But we typically cannot do much better than determining these contributions. Sep 18, 2023 at 0:44

I assumed that the increase in # of collisions is because of increased acceleration of each particle.

This is correct that the particles have to be accelerated to a larger velocity to exert more pressure on the walls of the container. The change in velocity is due to the change in the energy of the gas ($$dU$$) due to the heating from the external source that is raising the temperature of the gas in the container. From the first law of thermodynamics - for a fixed volume, we have $$dU = dQ \,,$$

where $$dQ$$ is the energy supplied from outside. Now, if we assume that this energy goes in accelerating the gas particles, then from the conservation of energy, we have

$$dU = \Delta {\rm K.E.}$$

where $$\Delta {\rm K.E}$$ is the change in the kinetic energy of the gas and can be written as $$\Delta {\rm K.E} = \frac{1}{2} \sum_{i = 1}^N m (v_{\rm f, i}^2 - v_{\rm in, i}^2) \,.$$

Above $$N$$ is the number of gas particles in the container, each having mass $$m$$ and $$v$$ is the final and initial velocities.

The origin of the force is microscopic in nature. This essentially comes from the heating of the walls on the container. Heating causes the particles in the walls of the container to move at a larger velocity than the particles in the gas (within the box). This energy is then transmitted to the gas by collisions.

• Hi! Thanks for answering! I wanted to connect this concept to newton's 2nd law. In short, what "force" propelled the particles to accelerate? I wanted to say temperature, but I doubt that it is a force. Thanks! Sep 18, 2023 at 0:32
• The true nature of the force would be electromagnetic in origin. Whatever is heating the container is first accelerating the air particles in the flame (if the fire is how your heating happening). These particles are then colliding with the wall of the container and transferring their momentum to it. How the particles in the flame are accelerated in the first place? These come from the chemical reactions of the material being burned which is eventually electromagnetic in nature.
– S.G
Sep 18, 2023 at 0:44

Part of the heat transferred to the molecules is converted into kinetic energy of the molecules.Then the molecules have more momentum so they exchange more momentum with the walls of the box.