When the total average kinetic energy of a gas, given by $ E_K = \frac{1}{2}mv^2 $, is incorporated into the previous equation, we obtain: $ P = \frac{2}{3} \frac{E_K}{V} $. Considering that the movement of the piston leads to a change in pressure within the gas column, and the work done on the gas by the piston is stored as internal energy, it follows that work corresponds to a change in the internal energy of the gas. The force required to compress the gas is $ P \times S $, so when the length of the gas column changes to $ v_l $, the work done is: $ W = -PSv_l $, with the work being positive when $ v_l $ is negative. Assuming all this work goes into increasing the average kinetic energy of the molecules, we can write: $ \Delta E = PSv_l $. However, the change in length $ v_l $ not only affects the kinetic energy $ E_K $ but also alters the pressure $ P $. Taking the differential, we get: $ \Delta P = \frac{2}{3S} \left( \frac{1}{l} \Delta E_K - \frac{v_l}{l^2} E_K \right) $, which simplifies to: $ \Delta P = \frac{2}{3S} \left( \frac{\Delta E_K}{l} - \frac{v_l}{l} \cdot \frac{2}{3} \frac{E_K}{l} \right) $. Substituting $ \Delta E_K = PSv_l $, the expression becomes: $ \Delta P = \frac{2}{3S} \left( -PSv_l \cdot \frac{1}{l} - P \cdot \frac{v_l}{l} \right) $. Since the cross-sectional area of the gas column is constant, the ratio $ \frac{v_l}{l} $ equals the relative volume change $ \frac{\Delta V}{V} $. Therefore: $ B_{JR} = -\frac{V \Delta P}{\Delta V} = \frac{5}{3}P $ (Note: $ B_{JR} $ represents the adiabatic gas elasticity modulus). In reality, this formula is only valid for ideal gases, not for all types of gases or even for air. This is because during the derivation, two key assumptions were made: first, that all work done on the gas during compression is used to increase its internal energy without any loss to the surroundings; second, that all the energy is converted into translational kinetic energy of the molecules, with none going into rotational or vibrational modes. While the first assumption holds true for acoustic vibrations in all gases, the second is only accurate for monatomic gases like helium, neon, or argon—molecules that behave like solid marbles. For polyatomic gases, such as air, part of the work done increases the internal rotational or vibrational energy of the molecules. As a result, the translational kinetic energy responsible for pressure changes is less than what the formula predicts. Thus, the adiabatic elasticity modulus for an ideal gas can be expressed as: $ B_{JR} = \gamma P $, where $ \gamma = \frac{5}{3} $ for monatomic gases.
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