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.

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

where $\displaystyle{U}$ is the internal energy.
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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

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

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.