Problem 2.1b

A First Course in String Theory


2.1 Exercises with units

(b) Explain the meaning of the unit K (degree kelvin) used for measuring temperatures, and explain its relation to the basic length, mass, and time units.


{\displaystyle {\frac {1}{T}}=\left({\frac {\partial S}{\partial U}}\right)_{V,N}},

where \displaystyle{U} is the internal energy.

The units of \displaystyle{k_B T} and \displaystyle{E} are the same.

\displaystyle{[k_B T] = [E]}

In other words, the Boltzmann constant \displaystyle{k_B} translates the temperature unit \displaystyle{K} to the language of energy unit \displaystyle{J}.

However, although the temperature unit \displaystyle{K} and the energy unit \displaystyle{J} have the relation

\displaystyle{k_B K = J},

just \displaystyle{k_B T} would not give the correct value of energy \displaystyle{E}, not to mention that we have not yet specified of which the energy \displaystyle{E} is.


For an ideal gas,

\displaystyle{pV=Nk_B T}

and the average translational kinetic energy is

{\displaystyle {\frac {1}{2}}m{\overline {v^{2}}}={\frac {3}{2}}k_BT}

for 3 degrees of freedom. In 3D space, if there are only translational motions, there are only 3 degrees of freedom.

In other words, just the value of {k_B T} itself gives no physical meaning. Instead, {\tfrac{1}{2}k_B T} can be interpreted as the average translational kinetic energy of the particles in an one dimensional space. Equivalently, \displaystyle{\tfrac{3}{2}k_BT} gives that in our three dimensional space.


Another main difference is that although energy is an extensive property, temperature is an intensive property.


We define the temperature unit Kelvin \displaystyle{K} by requiring the water triple point temperature,

\displaystyle{T_{tp} \equiv 273.16K}

Once this value is fixed, the Boltzmann constant \displaystyle{k_B} value can be estimated by using, for example, the ideal gas law

\displaystyle{pV = N k_B T},

because \displaystyle{k_B} always comes with \displaystyle{T}.

— Me@2020-02-16 11:14:24 AM



2020.02.16 Sunday (c) All rights reserved by ACHK