1.3: Dimensional Analysis

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

\( \newcommand{\upOmega}{\Omega}\)

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

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

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

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

Sometimes you can catch an error in the form of an equation or expression, or in the dimensions of a quantity used for a calculation, by checking for dimensional consistency. Here are some rules that must be satisfied:

• In this e-book the differential of a function, such as \(\df\), refers to an infinitesimal quantity. If one side of an equation is an infinitesimal quantity, the other side must also be. Thus, the equation \(\df = a\dx + b\dif y\) (where \(ax\) and \(by\) have the same dimensions as \(f\)) makes mathematical sense, but \(\df = ax+b\dif y\) does not.

  Derivatives, partial derivatives, and integrals have dimensions that we must take into account when determining the overall dimensions of an expression that includes them. For instance:

  • Some examples of applying these principles are given here using symbols described in Sec. 1.2.

    Example 1. Since the gas constant \(R\) may be expressed in units of J K\(^{-1}\) mol\(^{-1}\), it has dimensions of energy divided by thermodynamic temperature and amount. Thus, \(RT\) has dimensions of energy divided by amount, and \(nRT\) has dimensions of energy. The products \(RT\) and \(nRT\) appear frequently in thermodynamic expressions.

    Example 3. Find the dimensions of the constants \(a\) and \(b\) in the van der Waals equation \[ p = \frac {nRT}{V-nb} - \frac {n^{2}a} {V^2} \] Dimensional analysis tells us that, because \(nb\) is subtracted from \(V\), \(nb\) has dimensions of volume and therefore \(b\) has dimensions of volume/amount. Furthermore, since the right side of the equation is a difference of two terms, these terms have the same dimensions as the left side, which is pressure. Therefore, the second term \(n^{2}a/V^{2}\) has dimensions of pressure, and \(a\) has dimensions of pressure \(\times\) volume\(^{2}\) \(\times\) amount\(^{-2}\).

    Example 4. Consider an equation of the form \[ \Pd{\ln x}{T}{\!p} = \frac {y}{R} \] What are the SI units of \(y\)? \(\ln x\) is dimensionless, so the left side of the equation has the dimensions of \(1/T\), and its SI units are K\(^{-1}\). The SI units of the right side are therefore also K\(^{-1}\). Since \(R\) has the units J K\(^{-1}\) mol\(^{-1}\), the SI units of \(y\) are J K\(^{-2}\) mol\(^{-1}\).