# What is the relation of particle velocity, temperature, and reaction activation energy

So, this curiosity has arisen for a fun project I thought I'd tackle, where I'm attempting create a crude simulation of an internal combustion cylinder.

I wanted to explore the particle level interactions of a 4 step methane reaction, and see how they effect macroscopic properties like cylinder wall and force on the piston head. So the initial thought was simulate a set of particles modeled as hard spheres bouncing around, which I've found, are the general assumptions in kinetic theory. My background is with the macroscopic (engineering) and I'm trying to understand how these particle interactions of these relate to my existing macro knowledge.

I've read up on the Maxwell-Boltzmann distribution, which makes sense that for a collection of particles at a given temperature there is a probability of finding particles at a given speed in a small box.

$$f(v) = \sqrt{\frac{m}{2\pi KT}^{3}} 4\pi v^2 e^{\frac{-mv^2}{2kT}}$$

But how is temperature defined for an individual particle, is there such a thing? I was also looking into other definitions of temperature which don't seem to apply to individual particles such as:

$$1/T = \frac{\delta S}{\delta U}$$

If anything this even more confusing because, again I'm not sure how these terms are defined for individual particles. So is temperature some type of scalar property in addition to velocity information? My understanding is particle velocity and temperature are independent properties.

Finally, I was hoping to take some properties of my particle and on collision have some model to determine if a reaction would or wouldd not occur. How is activation energy related to velocity and temperature on a particle level, or can individual reactions not be modeled from a Newtonian stand point?

• If I understand this correctly, what you are interested in are combustion processes? In that case the single particle models are not going to go anywhere. You are looking at one of the more complicated problems in computational fluid dynamics, which means that you will have to understand thermodynamics and hydrodynamics at the same time. To find relevant literature, I would suggest a search for "combustion modeling", that will, at least, get you an overview of the technical terms and approaches. – CuriousOne Mar 4 '16 at 15:04
• You really want to look at collision theory: en.wikipedia.org/wiki/Collision_theory – Gert Mar 4 '16 at 15:11
• Yes, I'm essentially trying to model a combustion process which I've been learning about in great lecture series from princeton here. I've done some modeling using Fluent where all the species are continuous quantities but wanted to discrete approach a shot. – John Chabot Mar 4 '16 at 16:02
• I was actually looking at collision theory just before posting this question. If it is applicable to individual collisions, I wasn't how an individual particles properties would determine if a reaction could occur or not – John Chabot Mar 4 '16 at 16:04

Temperature is defined for the whole system and for an individual particle velocity is defined. If you want to simulate particles movement you should generate Maxwell-Boltzmann distribution for a given temperature. To generate the distribution function, you should take different values of the speed and calculate probability $f(v)$. Probability can be recalculated to the particle number.

The simplest way to calculate the probability p of the reaction is by using Arrenius law. $p = \exp(-\frac{E}{kT})$ Here E is the activation energy. If you want to calculate rates of the reaction you should also calculate number of the collisions.

Temperature can be defined as a function of mean particle velocity.

Thermodynamics treats gases, liquids, and solids as continuums, so here heat flows from higher temperatures to lower temperatures, and at Equilibrium there is no net transfer of energy.

Statistical Mechanics treats all matter as composed of atoms, Kinetic Theory looks at velocities and momenta, and Classical Statistical Mechanics looks at energies, both use Classical Statistical Distributions to calculate gross properties such as Temperature, Pressure, Entropy.

Quantum Statistical Mechanics does the same, using the Postulates of Quantum Theory, and is a whole new ball-game, … F-D and B-E distributions; but is probably the branch of Quantum Theory which has brought most benefit to humanity, leading to the development of lasers and semiconductor devices.

• Temperature is not the mean particle velocity, that would be the bulk flow velocity. Temperature can be derived from the average kinetic energy of a fluid in the bulk flow rest frame (but it can be derived in other ways too). – honeste_vivere Mar 16 '16 at 12:37