Temperature

The particles that make up an object can have ordered energy and disordered energy.  The kinetic energy of an object as a whole due to its motion with velocity v with respect to an observer is an example of ordered energy.  The kinetic energy of individual atoms, when they are randomly vibrating about their equilibrium position, is an example of disordered energy.  Thermal energy is disordered energy.  The temperature is a measure of this internal, disordered energy.

Definition:

The absolute temperature of any substance is proportional to the average kinetic energy associated with the random motion of the atoms or molecules that make up the substance.

In a gas, the individual atoms and molecules are moving randomly.  The absolute temperature T of the gas is proportional to the average translational kinetic energy of a gas atom or molecule, ½m<v2>.  In SI units, the proportional constant is (3/2)kB, where kB = 1.381*10-23 J/K or 1.381*10-23 Pa m3/K is called the Boltzmann constant

½m<v2> = (3/2)kBT

In a solid, the atoms can move randomly about their equilibrium positions.  In addition, the solid as a whole can move with a given velocity and have ordered kinetic energy.  Only the kinetic energy associated with the random motion of the atoms is proportional to the absolute temperature of the solid.

In ideal gases the disordered energy is all kinetic energy, in molecular gases and solids it is a combination of kinetic and potential energy.  If we model the atoms in a solid as being held together by tiny springs, then the random internal energy of each atom constantly switches between kinetic energy and elastic potential energy.

In classical physics, zero absolute temperature means zero kinetic energy associated with random motion.  The atoms in a substance do not move with respect to each other.  (The uncertainty principle in quantum mechanics requires that there is some zero-point energy.)   Room temperature is not close to absolute zero temperature.  At room temperature the atoms and molecules of all substances have random motion.

In SI units the scale of absolute temperature is Kelvin (K).  The Kelvin scale is identical to the Celsius (oC) scale, except it is shifted so that 0 degree Celsius equals 273.15 K.  We have

temperature in oC = temperature in K - 273.15.

To convert to temperature in Fahrenheit we can use

temperature in oF = (9/5) * temperature in oC + 32.

Problem:

Liquid nitrogen has a boiling point of -195.81 oC at atmospheric pressure.  Express this temperature in
(a) degrees Fahrenheit and
(b) Kelvin.

Solution:
(a) temperature in oF = (9/5) * temperature in oC + 32.
temperature in oF = [(9/5)(-195.81) + 32] oF = -320.5 oF.
(b) temperature in K = (-195.81+ 273.15) K = 77.34 K. 

Problem:

One of the hottest temperatures ever recorded on the surface of Earth was 134 oF in Death Valley, CA.
(a)  What is this temperature in oC?
(b)  What is this temperature in Kelvin?

Solution:
(a)  (5/9)*(temperature in oF - 32)= temperature in oC.
(5/9)*(134 - 32) oC = 56.67 oC.
(b)  temperature in oC + 273.15 = temperature in K.
(56.67 + 273.15) K = 329.82 K.

Problem:

(a)  At what temperature do the Fahrenheit and Celsius scales have the same numerical value?
(b)  At what temperature do the Fahrenheit and Kelvin scales have the same numerical value?

Solution:
(a) temperature in oF = (9/5) * temperature in oC + 32.
X = (9/5) * X + 32,  X - (9/5)X = 32,  -(4/5)X = 32, X = -5*32/4 = -40.
-40 oF = -40 oC.
(b)  temperature in oC = (5/9)*(temperature in oF - 32) = temperature in K - 273.15.
(5/9)*(temperature in oF - 32) + 273.15 = temperature in K.
(5/9)*(X - 32) + 273.15 = X,  (X - 32) + 491.67 = (9/5)X,   459.67 = (4/5)X,  X = 574.59.
574.59 oF = 574.59 K.