Equation of State of Pressure, Density, and Heat Capacity Ratio

This video presents a mechanical derivation of the equation of state of pressure, density, and heat capacity ratio. This equation is one of the three equations used to derive the acoustic wave equation. The equation of state for the compressibility relation is an equation that connects the pressure change and density change to the heat capacity ratio, gamma. In the equation, P-naught represents the average pressure, P is the absolute pressure, rho-naught is the average density, and rho is the absolute density. Gamma is the heat capacity ratio, equal to three alpha plus two, divided by three alpha. Where, alpha is defined as the total kinetic energy, Et plus Er, divided by the translational kinetic energy, Et. Based on this relationship, when the pressure increases, the density will also increase. The equation of state will be derived in this video based on the kinetic theory of gases. The equation of state can be derived from two equations: Equation A and Equation B. Combining Equations A and B and eliminating the common term gives the relationship between pressure change and volume change. Replacing volume change with density change, based on the conservation of mass, results in the equation of state. You might ask how to derive Equation A. The total kinetic energy, Ka, of a particle can be expressed as the sum of translational kinetic energy, Et, and rotational energy, Er. The translational kinetic energy, Et, equals half m-a times v squared; the rotational kinetic energy, Er, equals half I times theta dot squared. Where I-a is the moment of inertia and theta dot is the angular speed. Because pressure is only related to translational energy, Et, not rotational energy, Er, we want to relate the total kinetic energy, K-a, to translational kinetic energy, Et. To achieve this, we can define a variable, alpha, as the total kinetic energy divided by the translational energy. By this definition, we have K-a = alpha times Et. Replacing the translational kinetic energy, Et, with half m-a times v-c-a squared, we get K-a equals alpha times half ma times v-c-a squared. We know that the work done equals the energy increase of a closed system. The work done, W, is equal to negative pressure, P-nut, times delta V-a. The increase in total kinetic energy, delta K-a, can be obtained from the increase in the total kinetic energy that we have derived. From this, we obtain Equation A. Similarly, you might ask how to derive Equation B. Equation B is derived from the pressure vs. collision speed equation that relates the macroscopic pressure to the microscopic particle collision speed. The derivation of this equation is shown in another video. Taking the difference on both sides of the equation gives Equation B. In Equation A and Equation B, there is a common term, delta m-a times v-c-a squared. Eliminating this common term gives the relationship between pressure change and volume change. Rearrange the equation to have pressure on one side and volume on the other side. Based on the conservation of mass, the density times volume, V-a, is constant. Therefore, the derivative of this constant is zero. Substituting this equation into the relationship between pressure change and volume change relates the pressure change to the density change. Simplifying the equation using the heat capacity ratio, gamma, gives the final equation of state in terms of delta-P-nut and delta-rho-nut or P minus P-nut and Rho minus Rho-nut. In summary, the equation of state can be derived from Equation A and Equation B. Combining Equations A and B gives the relationship between pressure change and volume change. Replacing volume change with density change, based on the conservation of mass, gives the equation of state.
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