Reversible Adiabatic Processes

Basic Thermodynamics ~ J. Pohl ©

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Frictionless Adiabatic Process: IDEAL GAS

The development presented here is very important. Our first task is to describe the idealization, "frictionless," as it applies to an ideal gas event.

The friction we are discussing occurs at the boundary of the gas and also "inside the the air." There is friction of the bullet in the barrel also. Beginning level methods are incapable of dealing with friction. With the barrel, we just ignore friction. To ignore friction in the air we assume its pressure decreases as it expands, but does so without friction, uniformly throughout all of the air. This means we assume, p_B = p for the air for all stages of its expansion. With this change, we arrive at a simple differential equation for the air. The equation and its conditions are:

(9)(9)

The first step of solution is to use the gas equation to replace p with T and v. Then separate the variables and integrate.

(9)(9)

By the rules of natural logarithms, the expression above right can be cast as a process path for the frictionless adiabatic expansion (or compression of a gas). Our immediate result has T and v as variables. Substitutions using the gas equation with a workout in dealing with exponents yield an equation form with T and p and another with p and v as variables. These are included here for completeness.

()()

The left most form applies to our air pistol. Entering our numbers, the second temperature of the air is determined to be 267 K.

(13)(13)

It is now possible to solve for the exiting velocity of the bullet.

(14)(14)