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The chemical potential of a substance in a mixture, $\\mathrm{\\mu_x}$, is the contribution of that substance to the total Gibbs free energy, $\\mathrm{G}$, of the mixture. It is effectively the molar Gibbs free energy inherent to that substance under a particular set of conditions.

As a substance is formed, more of its chemical potential is introduced - and as a substance is consumed, its chemical potential is removed - from the total $\\mathrm{G}$ value. Since $\\mathrm{\\mu_x}$ is molar (i.e. it results from one mole of the substance), the change in $\\mathrm{G}$ when a substance is formed or consumed is proportional to the number of moles of the substance formed or consumed which we denote $\\mathrm{\\Delta n}$:

$\\mathrm{\\Delta G=\\Delta n\\cdot \\mu_x}$

Let's say that $\\mathrm{\\Delta n}$ moles of $\\mathrm{O_2}$ are used up in the reaction:

$\\mathrm{2SO_2\\space +\\space O_2\\space \\rightleftharpoons\\space 2SO_3}$.

Since $\\mathrm{\\Delta n}$ moles are consumed, we will re-express the change in moles of $\\mathrm{O_2}$ as $\\mathrm{-\\Delta n}$.

From the reaction, we can see that twice as much $\\mathrm{SO_2}$ is consumed, so the change in moles of $\\mathrm{SO_2}$ is $\\mathrm{-2\\Delta n}$.

Similarly, we can see that twice as much $\\mathrm{SO_3}$ is formed than $\\mathrm{O_2}$ consumed, but we must pay attention to the fact that that much $\\mathrm{SO_3}$ is being formed, so the change in moles of $\\mathrm{SO_3}$ is $\\mathrm{+2\\Delta n}$.

Knowing all this, we can express the net change in the Gibbs free energy of the mixture in terms of the chemical potentials of the three substances:

$\\mathrm{\\Delta G=-2\\Delta n\\mu_{_{SO_2}}-(1)\\Delta n\\mu_{_{O_2}}+2\\Delta n\\mu_{_{SO_3}}}$ .

Now, we have all three chemical potentials, so before we use this equation, we just need to calculate $\\mathrm{\\Delta n}$, i.e. the number of moles of $\\mathrm{O_2}$ consumed.

Note: You may have noticed that the chemical potentials we were given were expressed in $\\mathrm{kJ\\space mol^{-1}}$. We have no need to convert these to $\\mathrm{J\\space mol^{-1}}$, as the equation we are going to use them in only involves multiplication by constants, and addition; in this case we only need to make sure that the chemical potentials are all expressed in the same units, so that we can sum them together fairly; the only consequence being that our answer will also be scaled up by one thousand, so we will obtain a value for $\\mathrm{\\Delta G}$ already expressed in $\\mathrm{kJ}$, which we are asked for anyways.

We can calculate $\\mathrm{\\Delta n}$ from the mass of $\\mathrm{O_2}$ consumed, $\\mathrm{\\Delta m=\\var{m}\\space g}$, noting that the $\\mathrm{gFM}$ of $\\mathrm{O_2}$ is $\\mathrm{\\var{gfmo2}\\space g\\space mol^{-1}}$:

$\\mathrm{n=\\frac{m}{gFM}}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta n=\\frac{\\Delta m}{gFM}}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta n=\\frac{\\var{m}}{\\var{gfmo2}}=(\\var{sigformat(n,5)}\\cdots)\\space mol}$ .

Note: You should try to leave this value in your calculator for the next calculation, or note it down to at least the number of significant figures shown.

We can now use our equation to calculate $\\mathrm{\\Delta G}$ for the reaction (for ease, we will first factor $\\mathrm{\\Delta n}$ out in front):

$\\mathrm{\\Delta G=-2\\Delta n\\mu_{_{SO_2}}-\\Delta n\\mu_{_{O_2}}+2\\Delta n\\mu_{_{SO_3}}}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta G=\\Delta n(-2\\mu_{_{SO_2}}-\\mu_{_{O_2}}+2\\mu_{_{SO_3}})}$

$\\mathrm{\\Rightarrow\\space\\space\\space \\Delta G=(\\var{sigformat(n,5)}\\cdots)\\times([-2\\times\\var{gso2}]-0+[2\\times\\var{gso3}])=\\var{sigformat(gibbs,4)}\\space kJ}$ .

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Calculate $\\mathrm{\\Delta G}$ for the reaction:

$\\mathrm{2SO_2\\space +\\space O_2\\space \\rightleftharpoons\\space 2SO_3}$

when $\\mathrm{\\var{m}\\space g}$ of $\\mathrm{O_2}$ are consumed, given the measured chemical potentials $\\mathrm{\\mu_{_{SO_2}}=\\var{gso2}\\space kJ\\space mol^{-1}}$, $\\mathrm{\\mu_{_{O_2}}=\\var{go2}\\space kJ\\space mol^{-1}}$, and $\\mathrm{\\mu_{_{SO_3}}=\\var{gso3}\\space kJ\\space mol^{-1}}$ (you may assume these values remain constant during the specified consumption of oxygen):

[[0]] $\\mathrm{kJ}$

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