As a high school calculus and physics student interested in aerospace engineering, I was trying to understand the basics of how a rocket works and couldn’t find an equation or any analysis online on what amount of the energy released by the fuel, once combusted, turns into heat versus how much propels the gas forward and pressurizes the combustion chamber, and seeing how those values are essential to calculate thrust (total pressure and total temperature), I’ve been lost by trying to find how to calculate these values. It is possibly I am understanding how combustion works wrong, but I haven’t been able to find anything that has helped me learn how to calculate these things theoretically instead of experimentally. Any help is greatly appreciated!
Sutton,4th Edition, page 7:
The energy from a high-pressure combustion reaction of propellant chemicals, usually a fuel and an oxidizing chemical, permits the heating of reaction product gases to very high temperatures (4500 to 7500 deg F). These gases subsequently are expanded in a nozzle and accelerated to high velocity.
So it's not "heat versus what propels the gas forward". The heat is what propels the gas forward and pressurizes the combustion chamber.
You're basically asking how to calculate the combustion chamber conditions and that is a long answer, probably not suitable for Stack Exchange. Instead I suggest you take a look at the version of Sutton available online, although inferior in many people's judgment to earlier editions, the basics are in there. Study up on that and then come back with specific questions.
Here's the big picture on how to do it from the 4th edition (page 181):
The combustion chamber conditions (such as chamber temperature and gas composition) can be calculated by using the conditions of mass balance (-), the pressure balance (-), several chemical equilibrium conditions (-), and the energy balance (-), and by simultaneously solving these equations....The unknowns in these equations are the chamber temperature Tc and the molar fractions nj of each of the β constituents in the reaction product gases; thus the number of independent equations must equal β+1.
(-) I removed equation reference here because it's irrelevant to the point and doesn't apply to the book available online.
edit: @JohnRennie's Physics SE answer does a much better job explaining this than I ever could; after reading this, reading that would be time well spent, as is time spent reading any of his answers.
@OrganicMarble's answer is excellent and gives you a resource to read further. I'll add an answer that might be helpful intuitively.
Expansion of gases generally leads to cooling. We know that from the Ideal Gas Law and from the fact that refrigerators and air conditioners work.
So what happens to the kinetic energy of all the gas molecules when the gas cools?
If you pop a balloon in a vacuum, the gas cools, and the kinetic energy associated with the original temperature remains present, but changes from thermal energy to kinetic energy of the directed, radial expansion.
Expansion provides a preferred direction for the molecular motion. Instead of completely random motion, the atoms move preferentially in the direction of expansion. Temperature is a measure of the random motion, so the kinetic energy of any ordered motion of a region of gas no longer contributes to the thermal energy or temperature.
Ultimately the kinetic energy in a contained expansion like a refrigerator or air conditioner or combustion engine cylinder does mechanical work on the expansion cell. This answer might be helpful to read at this point.
The balloon popped in a vacuum is a helpful analogy to what happens in a rocket nozzle. The controlled, optimized expansion converts the random motion of the hot gas to the directed motion of the exhaust moving backwards.