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This may fall a little under chemistry processes but I felt it has enough pertaining to aerospace to put it here. Basically, I'm in the process of attempting to develop a way to derive $I_{sp}$ as pertaining to rocket engines rather than rely on charted information.

Since specific impulse is essentially the exhaust gas velocity working against gravitational force,

$$ I_{sp} = \frac{v_e}{g_0} $$

...it stands to reason that the ideal exhaust gas velocity equation can be substituted in here, giving something like

$$ I_{sp} = \frac{\sqrt{\frac{TR}{M}\cdot\frac{2\gamma}{\gamma-1}\cdot(1-\frac{\rho_e}{\rho}^{\frac{\gamma - 1}{\gamma}})}}{g_0} $$

The obvious problem here is that this is the ideal exhaust gas velocity, so this is a sort of "perfect-universe" $I_{sp}$. Because most rocket engines use either hydrocarbon or hydrogen/oxygen propellants, water vapor is a major fraction of this exhaust. And as is usually taught early on in chemistry courses, water vapor is a textbook failure of ideal gas behaviors because of its intermolecular forces.

So my question is - is there a "real" specific impulse formula; something like the van der Waals equation for this application?

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  • $\begingroup$ Considering the number of variables unaccounted for in measurement of exhaust speed - imperfect flow, condensation, many more - the most accurate practical formula would be $ I_{sp} = {\Delta v \over g_0}{ m_1 \over m_0} $ derived from Tsiolkovsky Equation. These values can be accurately measured and there are no hidden variables, simplifications or other dependencies in this equation. $\endgroup$
    – SF.
    Commented Sep 25, 2015 at 22:21

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There is no general formula for isp that would provide accurate values all the propellant combinations and nozzle expansion ratios one might use. The NASA Chemical Equilibrium (CEA) program can provide the information you are looking for and would be an answer to your question. It is useful for any of the common propellant systems and many exotic propellant systems, e.g., fluorine, metals, etc.

Extracted from nasa CEA webpage:

CEA is a program which calculates chemical equilibrium product concentrations from any set of reactants and determines thermodynamic and transport properties for the product mixture. Built-in applications include calculation of theoretical rocket performance (isp at any nozzle expansion ratio), Chapman-Jouguet detonation parameters, shock tube parameters, and combustion properties.

The code in fortran is freely available (check web site). This code is the "bible" for the rocket industry. I used this 50 years ago while working on the Apollo program and working on other rocket systems.

The program is very fast. You could set CEA up to be a call from your application or you could run a bunch of cases with the propellants you are most interested in and call a curvefit to the data for the specific situations you are looking at.

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  • $\begingroup$ I've thought of many things I could say about the CEA program, but none suffices but "Wow". There's a lot I'm going to learn from this source code. $\endgroup$
    – ecfedele
    Commented Apr 28, 2015 at 21:25

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