If the total average kinetic energy of EK:EK=12mv2 is introduced into the above formula, it is obtained: P=23SEKl Considering the movement of the piston causes the pressure in the gas column to change, and the work done by the piston on the gas is stored in the gas. Therefore, work represents the change of the internal energy of the gas. The force required to compress the gas is PS, so when the length of the gas column changes to vl, the work done is: vW = -PSvl, and when vl is negative, the work is positive. It is assumed that this work is all used to increase the average kinetic energy of the molecule, so there is vE=PSvl. However, the length change vl is accompanied not only by the change of EK but also by the change of pressure P; the differential is: vP=23S (1lvEK) -vll2EK)vP=23SlvEK-vll(23SEKl), the above formula becomes vP=23Sl(-PSvl)-pvll because the cross-sectional area of ​​the gas column is assumed to be constant, the value of vl/l is equal to the relative change of volume. vV/V. Therefore: BJR=-VvPvV=53P (Note: BJR is the adiabatic gas elastic modulus) In fact, the formula is only true for the ideal gas, not for all gases, nor for air.
Because in the process of deriving this formula, the following assumptions are made: First, all the work done on the gas during compression is used to increase the energy of the gas, and no energy is lost to the surrounding matter; second, these are obtained by the gas. All of the energy is used to increase the average kinetic energy of the molecule without using a portion of the energy to increase the internal kinetic energy of the molecule. For the acoustic vibration of all gases, the first condition is met. However, the second condition applies only to molecules that are actually like a solid marble ball, especially like a single atomic gas such as He, Ne, or Ar. As for other gases, including air, some of the work done on the gas causes the internal rotation or vibrational energy of the molecule to change. Therefore, for a certain volume change, the amount of change in the translational kinetic energy that determines the pressure in the equation is smaller than the value given by the above calculation. Therefore, the elastic modulus of an ideal gas in adiabatic vibration can be expressed as: BJR=CP,1
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