# Chapter 6: The Third Law and Cryogenics

$$\newcommand{\tx}[1]{\text{#1}} % text in math mode$$
$$\newcommand{\subs}[1]{_{\text{#1}}} % subscript text$$
$$\newcommand{\sups}[1]{^{\text{#1}}} % superscript text$$
$$\newcommand{\st}{^\circ} % standard state symbol$$
$$\newcommand{\id}{^{\text{id}}} % ideal$$
$$\newcommand{\rf}{^{\text{ref}}} % reference state$$
$$\newcommand{\units}[1]{\mbox{\thinspace#1}}$$
$$\newcommand{\K}{\units{K}} % kelvins$$

$$\newcommand{\degC}{^\circ\text{C}}} % degrees Celsius$$

$$\newcommand{\br}{\units{bar}} % bar (\bar is already defined)$$
$$\newcommand{\Pa}{\units{Pa}}$$
$$\newcommand{\mol}{\units{mol}} % mole$$
$$\newcommand{\V}{\units{V}} % volts$$
$$\newcommand{\timesten}[1]{\mbox{\,\times\,10^{#1}}}$$
$$\newcommand{\per}{^{-1}} % minus one power$$
$$\newcommand{\m}{_{\text{m}}} % subscript m for molar quantity$$
$$\newcommand{\CVm}{C_{V,\text{m}}} % molar heat capacity at const.V$$
$$\newcommand{\Cpm}{C_{p,\text{m}}} % molar heat capacity at const.p$$
$$\newcommand{\kT}{\kappa_T} % isothermal compressibility$$
$$\newcommand{\A}{_{\text{A}}} % subscript A for solvent or state A$$
$$\newcommand{\B}{_{\text{B}}} % subscript B for solute or state B$$
$$\newcommand{\bd}{_{\text{b}}} % subscript b for boundary or boiling point$$
$$\newcommand{\C}{_{\text{C}}} % subscript C$$
$$\newcommand{\f}{_{\text{f}}} % subscript f for freezing point$$
$$\newcommand{\mA}{_{\text{m},\text{A}}} % subscript m,A (m=molar)$$
$$\newcommand{\mB}{_{\text{m},\text{B}}} % subscript m,B (m=molar)$$
$$\newcommand{\mi}{_{\text{m},i}} % subscript m,i (m=molar)$$
$$\newcommand{\fA}{_{\text{f},\text{A}}} % subscript f,A (for fr. pt.)$$
$$\newcommand{\fB}{_{\text{f},\text{B}}} % subscript f,B (for fr. pt.)$$
$$\newcommand{\xbB}{_{x,\text{B}}} % x basis, B$$
$$\newcommand{\xbC}{_{x,\text{C}}} % x basis, C$$
$$\newcommand{\cbB}{_{c,\text{B}}} % c basis, B$$
$$\newcommand{\mbB}{_{m,\text{B}}} % m basis, B$$
$$\newcommand{\kHi}{k_{\text{H},i}} % Henry's law constant, x basis, i$$
$$\newcommand{\kHB}{k_{\text{H,B}}} % Henry's law constant, x basis, B$$
$$\newcommand{\arrow}{\,\rightarrow\,} % right arrow with extra spaces$$
$$\newcommand{\arrows}{\,\rightleftharpoons\,} % double arrows with extra spaces$$
$$\newcommand{\ra}{\rightarrow} % right arrow (can be used in text mode)$$
$$\newcommand{\eq}{\subs{eq}} % equilibrium state$$
$$\newcommand{\onehalf}{\textstyle\frac{1}{2}\D} % small 1/2 for display equation$$
$$\newcommand{\sys}{\subs{sys}} % system property$$
$$\newcommand{\sur}{\sups{sur}} % surroundings$$
$$\renewcommand{\in}{\sups{int}} % internal$$
$$\newcommand{\lab}{\subs{lab}} % lab frame$$
$$\newcommand{\cm}{\subs{cm}} % center of mass$$
$$\newcommand{\rev}{\subs{rev}} % reversible$$
$$\newcommand{\irr}{\subs{irr}} % irreversible$$
$$\newcommand{\fric}{\subs{fric}} % friction$$
$$\newcommand{\diss}{\subs{diss}} % dissipation$$
$$\newcommand{\el}{\subs{el}} % electrical$$
$$\newcommand{\cell}{\subs{cell}} % cell$$
$$\newcommand{\As}{A\subs{s}} % surface area$$
$$\newcommand{\E}{^\mathsf{E}} % excess quantity (superscript)$$
$$\newcommand{\allni}{\{n_i \}} % set of all n_i$$
$$\newcommand{\sol}{\hspace{-.1em}\tx{(sol)}}$$
$$\newcommand{\solmB}{\tx{(sol,\,m\B)}}$$
$$\newcommand{\dil}{\tx{(dil)}}$$
$$\newcommand{\sln}{\tx{(sln)}}$$
$$\newcommand{\mix}{\tx{(mix)}}$$
$$\newcommand{\rxn}{\tx{(rxn)}}$$
$$\newcommand{\expt}{\tx{(expt)}}$$
$$\newcommand{\solid}{\tx{(s)}}$$
$$\newcommand{\liquid}{\tx{(l)}}$$
$$\newcommand{\gas}{\tx{(g)}}$$
$$\newcommand{\pha}{\alpha} % phase alpha$$
$$\newcommand{\phb}{\beta} % phase beta$$
$$\newcommand{\phg}{\gamma} % phase gamma$$
$$\newcommand{\aph}{^{\alpha}} % alpha phase superscript$$
$$\newcommand{\bph}{^{\beta}} % beta phase superscript$$
$$\newcommand{\gph}{^{\gamma}} % gamma phase superscript$$
$$\newcommand{\aphp}{^{\alpha'}} % alpha prime phase superscript$$
$$\newcommand{\bphp}{^{\beta'}} % beta prime phase superscript$$
$$\newcommand{\gphp}{^{\gamma'}} % gamma prime phase superscript$$
$$\newcommand{\apht}{\small\aph} % alpha phase tiny superscript$$
$$\newcommand{\bpht}{\small\bph} % beta phase tiny superscript$$
$$\newcommand{\gpht}{\small\gph} % gamma phase tiny superscript$$

$$\newcommand{\upOmega}{\Omega}$$

$$\newcommand{\dif}{\mathop{}\!\mathrm{d}} % roman d in math mode, preceded by space$$
$$\newcommand{\Dif}{\mathop{}\!\mathrm{D}} % roman D in math mode, preceded by space$$
$$\newcommand{\df}{\dif\hspace{0.05em} f} % df$$

$$\newcommand{\dBar}{\mathop{}\!\mathrm{d}\hspace-.3em\raise1.05ex{\Rule{.8ex}{.125ex}{0ex}}} % inexact differential$$
$$\newcommand{\dq}{\dBar q} % heat differential$$
$$\newcommand{\dw}{\dBar w} % work differential$$
$$\newcommand{\dQ}{\dBar Q} % infinitesimal charge$$
$$\newcommand{\dx}{\dif\hspace{0.05em} x} % dx$$
$$\newcommand{\dt}{\dif\hspace{0.05em} t} % dt$$
$$\newcommand{\difp}{\dif\hspace{0.05em} p} % dp$$
$$\newcommand{\Del}{\Delta}$$
$$\newcommand{\Delsub}[1]{\Delta_{\text{#1}}}$$
$$\newcommand{\pd}[3]{(\partial #1 / \partial #2 )_{#3}} % \pd{}{}{} - partial derivative, one line$$
$$\newcommand{\Pd}[3]{\left( \dfrac {\partial #1} {\partial #2}\right)_{#3}} % Pd{}{}{} - Partial derivative, built-up$$
$$\newcommand{\bpd}[3]{[ \partial #1 / \partial #2 ]_{#3}}$$
$$\newcommand{\bPd}[3]{\left[ \dfrac {\partial #1} {\partial #2}\right]_{#3}}$$
$$\newcommand{\dotprod}{\small\bullet}$$
$$\newcommand{\fug}{f} % fugacity$$
$$\newcommand{\g}{\gamma} % solute activity coefficient, or gamma in general$$
$$\newcommand{\G}{\varGamma} % activity coefficient of a reference state (pressure factor)$$
$$\newcommand{\ecp}{\widetilde{\mu}} % electrochemical or total potential$$
$$\newcommand{\Eeq}{E\subs{cell, eq}} % equilibrium cell potential$$
$$\newcommand{\Ej}{E\subs{j}} % liquid junction potential$$
$$\newcommand{\mue}{\mu\subs{e}} % electron chemical potential$$
$$\newcommand{\defn}{\,\stackrel{\mathrm{def}}{=}\,} % "equal by definition" symbol$$

$$\newcommand{\D}{\displaystyle} % for a line in built-up$$
$$\newcommand{\s}{\smash[b]} % use in equations with conditions of validity$$
$$\newcommand{\cond}[1]{\\[-2.5pt]{}\tag*{#1}}$$
$$\newcommand{\nextcond}[1]{\\[-5pt]{}\tag*{#1}}$$
$$\newcommand{\R}{8.3145\units{J\,K\per\,mol\per}} % gas constant value$$
$$\newcommand{\Rsix}{8.31447\units{J\,K\per\,mol\per}} % gas constant value - 6 sig figs$$

$$\newcommand{\jn}{\hspace3pt\lower.3ex{\Rule{.6pt}{2ex}{0ex}}\hspace3pt}$$
$$\newcommand{\ljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}} \hspace3pt}$$
$$\newcommand{\lljn}{\hspace3pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace1.4pt\lower.3ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise.45ex{\Rule{.6pt}{.5ex}{0ex}}\hspace-.6pt\raise1.2ex{\Rule{.6pt}{.5ex}{0ex}}\hspace3pt}$$

The third law of thermodynamics concerns the entropy of perfectly-ordered crystals at zero kelvins.

When a chemical reaction or phase transition is studied at low temperatures, and all substances are pure crystals presumed to be perfectly ordered, the entropy change is found to approach zero as the temperature approaches zero kelvins: \begin{gather} \s{ \lim_{T\!\ra 0} \Del S=0 } \tag{6.0.1} \cond{(pure, perfectly-ordered crystals)} \end{gather} Equation 6.0.1 is the mathematical statement of the Nernst heat theorem or third law of thermodynamics.  It is true in general only if each reactant and product is a pure crystal with identical unit cells arranged in perfect spatial order.

Nernst preferred to avoid the use of the entropy function and to use in its place the partial derivative $$-\pd{A}{T}{V}$$ (Eq. 5.4.9).  The original 1906 version of his heat theorem was in the form $$\lim_{T\!\ra 0} \pd{\Del A}{T}{V}{=}0$$ (William H. Cropper, J. Chem. Educ., 64, 3–8, 1987).