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This question [ Chemical potential in Thermodynamics ] turns out have most of the answer. It wasn't tagged with the 'chemical-potential' label so I didn't see it when I was asking. Anyway, I'll re-state it as it applies to this question: chemical potential is the energy added per particle if entropy and volume are kept constant. The real Fundamental ...


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Take an area element $\mathrm{d}S$ perpendicular to the $x$-axis. How much energy is transported through this in unit time? Well, the flow rate is $v_x\mathrm{d}S$, and the energy density is $\frac{1}{2}\rho v^2 + \rho \varepsilon$, where $\varepsilon$ is internal energy. So multiplying these we have the energy transportation rate through $\mathrm{d}S$, ...



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