1.5: The Boltzmann Constant
- Page ID
- 556214
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)\[ \frac{N_i}{N} = \frac{e^{-\beta (\epsilon_i - \epsilon_0)}}{\displaystyle \sum_i e^{-\beta (\epsilon_i - \epsilon_0)}} \]
Now that we have the Boltzmann equation, the question becomes - how to make sense of it? What is the physical meaning of this mysterious \( \beta \) parameter that has suddenly appeared and is somehow related to enforcing constraints on the total energy of the system?
This is best investigated by means of a thought experiment...
Consider the problem of determining how 500 molecules are distributed among equally-spaced energy levels \( (\epsilon_i - \epsilon_0) = 0, 1, 2, 3, ... \) and how this distribution depends on the value of the parameter \( \beta = 1, 2 \) or \( 5 \).
Strategy: Compute Boltzmann factors \( e^{-\beta (\epsilon_i - \epsilon_0)} \) for each choice of \( \beta \) and \( (\epsilon_i - \epsilon_0) \), until the Boltzmann factors become negligible. Sum up all the Boltzmann factors for each \( \beta \) value and divide each Boltzmann factor by the sum. Multiply each of these values by \( N = 500 \).
Result:
| \( i \) | \( \beta = 1 \) | \( \beta = 2 \) | \( \beta = 5 \) |
|---|---|---|---|
| 0 | 316 | 432 | 497 |
| 1 | 116 | 59 | 3 |
| 2 | 43 | 8 | 0 |
| 3 | 16 | 1 | 0 |
| 4 | 2 | 0 | 0 |
| 5 | 1 | 0 | 0 |
| 6 | 0 | 0 | 0 |
Summarising the outcome of this numerical thought experiment, we find that as \( \beta \) increases, the population of particles occupying higher energy levels decreases, and so too does the total thermal energy of the system.
This is exactly the opposite of the behaviour we would expect to see with increasing temperature, i.e. \[\beta \propto \frac{1}{T} \]
Another way of expressing this relationship is using a constant of proportionality:
\[ \beta = \frac{1}{k_B T} \]
A precise value for this constant of proportionality can be derived by calculating the translational energy of an ideal monoatomic gas using both a statistical approach and from classical thermodynamics. However, this result turns out to be completely general, and this fundamental physical constant is referred to as the Boltzmann constant.
\[ k_\mathrm{B} = 1.3806503 \times 10^{-23}\ \mathrm{J}\ \mathrm{K}^{-1} \]
It is also now clear that temperature is a unique thermodynamic parameter that governs the most probable populations of quantum states when a system of identical but distinguishable non-interacting quantum particles is at thermal equilibrium. This is a very important result, as it provides the first example of relating atomic level knowledge of quantum states to macroscopic properties of chemical systems.