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# Equations for Temperature Limits

by Ron Kurtus (revised 11 November 2014)

The lower and upper temperature limits can be approached but not physically reached. There is a relationship between kinetic energy, speed of the particles and temperature. Absolute zero is the coldest possible temperature. The limit for the highest temperature is when the particles reach the speed of light.

Questions you may have include:

• What is relationship between kinetic energy, speed and temperature?
• What happens when a material is heated?
• What is the upper temperature limit?

This lesson will answer those questions. Useful tool: Units Conversion

## Relationships

There are equations that determine the relationship between kinetic energy of an ideal gas, temperature and velocity of the atoms or molecules in the gas.

Note: An ideal gas is a theoretical gas composed of randomly-moving point particles that interact only through elastic collisions. It is useful in determining simple equations, as opposed to the highly complex ones of the real world.

### Kinetic energy and temperature

The relationship between the kinetic energy of the molecules or atoms in an ideal gas and temperature is:

KE = 2kT/3

where

• KE = the kinetic energy of particles in an ideal gas in joules (J)
• k = Boltzmann's constant (a number that relates energy and temperature)
k = 1.38*10−23 joule/kelvin
• T = temperature in degrees kelvin (K)

Kinetic energy-temperature relationship equations for real-world gases, liquids and solids are too complex to work with at this level of study.

### Kinetic energy and velocity

The kinetic energy of a moving mass of particles is:

KE = ½mv²

where

• KE = kinetic energy in joules or kg-m²/s²
• m = mass in kilograms (kg)
• v = velocity in meters/second (m/s)
• = velocity squared or v*v in m²/s²
• ½mv² is ½ times m times

### Temperature and velocity

You can find the relationship between the temperature and the velocity of the particles in an ideal gas.

Since KE = 2kT/3 and KE = ½mv², you can substitute for KE to get 2kT/3 = ½mv². Then, you can multiply by 3 and divide by 2k to get:

T = 3mv²/4k

where

• T is measured in degrees kelvin (K)
• m is the mass of in kilograms
• v is in meters/second
• k is in joule/kelvin or kg-m²/s²-kelvin

## Absolute zero

It can easily be seen from T = 3mv²/4k that when T = 0 kelvin, the velocity of the particles v = 0. Thus the kinetic energy due to linear movement is zero. But the atoms still possess spin, which means they still have some energy.

Another fact is that the equation is really an approximation, since we are dealing with an ideal gas. A real-world gas would not be able to reach T = 0.

## Temperature and the speed of light limit

The greatest temperature possible is limited by how fast its atoms can travel. The upper limit that anything can travel is at the speed of light.

Although kinetic energy is KE = ½mv², the limiting energy is defined by Einstein's Theory of Relativity equation

E = mc²

where;

• m = the resting mass
• = the speed of light (c) squared

Thus, in theory, the highest possible temperature is defined by:

T = 3mc²/2k

You can calculate that temperature by substituting the appropriate values. This equation may not fit into the Theory of Relativity, since the mass of a particle increases dramatically as the particle approaches the speed of light. But, at the very least, it is an interesting exercise.

## Summary

The lower and upper temperature limits can be approached but not physically reached. The relationship between kinetic energy, speed of the particles and temperature determines that value of absolute zero and the limit for the highest possible temperature.

## Resources and references

Ron Kurtus' Credentials

### Websites

Kinetic Temperature - HyperPhysics

Physics Resources

### Books

Do you have any questions, comments, or opinions on this subject? If so, send an email with your feedback. I will try to get back to you as soon as possible.

## Students and researchers

www.school-for-champions.com/science/temperature_limits_equations.htm

Please include it as a reference in your report, document, or thesis.

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