Equations for the lower and upper limits of temperature - Succeed in Physical Science. Also refer to physics, motion, thermometer, heat transfer, kinetic energy, absolute zero, quantum mechanics, speed of light, electromagnetic waves, gravity, Theory of Relativity, Einstein, Ron Kurtus, School for Champions. Copyright © Restrictions
Equations for Temperature Limits
by Ron Kurtus (5 December 2006)
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. There is a mini-quiz near the end of the lesson.
Useful tools: Metric-English Conversion | Scientific Calculator.
Relationships
Kinetic energy and temperature
The relationship between the kinetic energy of the molecules or atoms in an ideal gas is
KE = 2kT/3
where:
- KE = the kinetic energy of particles in an ideal gas
- k = Boltzmann's constant (a number that relates energy and temperature)
- T = temperature in degrees Kelvin
k = 1.38*10-23 joule/kelvin
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
- v = velocity in meters/second
- v² = velocity squared or v*v in m²/s²
- ½mv² is ½ times m times v²
Temperature and velocity
We can find the relationship between temperature and the velocity of its particles
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 Kelvin
- m is 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
- c² = 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.
Surpass your limitations
Resources
The following resources provide information on this subject:
Websites
Kinetic Temperature - HyperPhysics
Books
Top-rated books on Temperature
Top-rated books on Absolute Zero
Mini-quiz to check your understanding
1. Why is an ideal gas used for temperature limit equations?
2. What does Boltzmann's constant do?
3. Why is can't the maximum temperature be reached?
If you got all three correct, you are on your way to becoming a Champion in Physical Science. If you had problems, you had better look over the material again.
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Equations for Temperature Limits