1.14.72: Variables: Independent and Dependent
- Page ID
- 391700
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)A colleague has filled a flask with water and asks us by phone to estimate the volume of water in the flask. Clearly this is an impossible task but our colleague offers further information. In answer to the first question, our colleague informs us that there are 2 moles of water in the flask. Immediately we suggest that the volume of water is \(36 \mathrm{cm}^{3}\). Not good enough! Our colleague demands a more precise estimate. We know that the volume of water(\(\ell\)) depends on temperature and pressure and so request new this information. We are told that the temperature is \(298.2 \mathrm{~K}\) and the pressure is \(101325 \mathrm{~N m}^{-2}\). We summarize this information in the following form.
\[\mathrm{V}=\mathrm{V}\left[298.2 \mathrm{~K} ; 101325 \mathrm{~N} \mathrm{~m} \mathrm{~m}^{-2} ;(\ell) ; 2 \text { moles }\right] \nonumber \]
Our colleague offers further information such as the vapor pressure and heat capacity of water(\(\ell\)) under these conditions. But we decline this offer on the grounds that no further information is required. We know that having defined the variables in the square brackets [.....], a unique volume \(\mathrm{V}\) is defined. We may not immediately know the actual volume but given a little time in a scientific library we will be in a position to report volume \(\mathrm{V}\).
The variables in the square brackets are the INDEPENDENT VARIABLES [1]. For a system containing one chemical substance we define the volume as follows.
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}\right] \nonumber \]
The term independent means that within limits [1] we can change \(\mathrm{T}\) independently of the pressure and \(\mathrm{n}_{1}\); change \(\mathrm{p}\) independently of \(\mathrm{T}\) and \(\mathrm{n}_{1}\); change \(\mathrm{n}_{1}\) independently of \(\mathrm{T}\) and \(\mathrm{p}\). There are some restrictions in our choice of independent variables. At least one variable must define the amount of all chemical substances in the system and one variable must define the 'hotness' of the system.
The molar volume of liquid chemical substance 1 at the specified temperature and pressure, \(V_{1}^{*}(\ell)\) is obtained from equation (b) by fixing \(\mathrm{n}_{1}\) at 1 mol. Thus
\[\mathrm{V}_{1}^{*}(\ell)=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}=1 \mathrm{~mol}\right] \nonumber \]
If the composition of a given closed system is specified in terms of the amounts of two chemical substances, 1 and 2, four independent variables \(\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right]\) define the independent variable \(\mathrm{V}\) [2]. Thus
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2}\right] \nonumber \]
Actually there is merit in writing equation (d) in terms of three intensive variables which in turn defines the molar volume \(\mathrm{V}_{\mathrm{m}}\) of the binary system at given mole fraction \(x_{1}=1-x_{2}\). Thus
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}\right] \nonumber \]
For a system containing i - chemical substances where the amounts can be independently varied, the dependent extensive variable \(\mathrm{V}\) is defined by equation (f).
\[\mathrm{V}=\mathrm{V}\left[\mathrm{T}, \mathrm{p}, \mathrm{n}_{1}, \mathrm{n}_{2} \ldots \ldots \ldots, \mathrm{n}_{\mathrm{i}}\right] \nonumber \]
Similarly the dependent intensive variable \(\mathrm{V}_{\mathrm{m}}\) is defined by equation (g).
\[\mathrm{V}_{\mathrm{m}}=\mathrm{V}_{\mathrm{m}}\left[\mathrm{T}, \mathrm{p}, \mathrm{x}_{1}, \mathrm{x}_{2} \ldots \ldots \ldots, \mathrm{x}_{\mathrm{i}-1}\right] \nonumber \]
Footnotes
[1] The phrase 'independent variable' is important. With reference to the properties of an aqueous solution containing ethanoic acid, the number of components for such a solution is 2, amount of water and amount of ethanoic acid. The actual amounts of ethanoic acid, water, ethanoate and hydrogen ions are determined by an equilibrium constant which is an intrinsic property of this system at given \(\mathrm{T}\) and \(\mathrm{p}\). From the point of the Phase Rule [2], the number of components equals 2. For the same reason when we consider the volume of a system containing only \(\mathrm{n}_{j}\) moles of water we disregard evidence that water partly self-dissociates into \(\mathrm{H}^{+} (\mathrm{aq})\) and \(\mathrm{OH}^{-} (\mathrm{aq})\).
[2] In terms of the Phase Rule, for two components (\(\mathrm{C} = 2\)) and one phase (\(\mathrm{P} = 1\)), the number of degrees of freedom \(\mathrm{F}\) equals 3. These degrees of freedom refer to a set of intensive variables. Hence, for a solution where substance 1 is the solvent and substance 2 is the solute, the system is defined by specifying the three (intensive) degrees of freedom, \(\mathrm{T}, \mathrm{~p}\) and, for example, solute molality.