CLASSICAL MECHANICS & THERMODYNAMICS - Chapter 9, Exercise 3 Solution ========================================================== Conceptual: Elastic Collisions and Why Bernoulli's Theory Was Doubted PROBLEM ------- Explain, using Chapter 4's own definition of an elastic collision and conservation of momentum, why kinetic theory's explanation of gas pressure genuinely required molecular collisions to be perfectly elastic - and why this specific requirement was one real reason Bernoulli's 1738 proposal met with skepticism for over a century. SOLUTION -------- Chapter 4 defined an elastic collision as one where total kinetic energy is conserved (not just momentum, which is conserved in EVERY collision, elastic or not). In an inelastic collision, by contrast, some kinetic energy is lost to heat, sound, or deformation with each impact. Kinetic theory's explanation of gas pressure depends on molecules colliding constantly - with each other and with the container's own walls - essentially forever, with no external energy source topping them back up. If these collisions were inelastic even to a small degree, each individual collision would drain away a little of the gas's own kinetic energy, converting it into some other form. Given the astronomical number of molecular collisions happening every second in even a small volume of gas, any real energy loss per collision - however tiny - would compound catastrophically fast, and the gas's molecules would rapidly slow to a stop, extinguishing both the pressure and the temperature kinetic theory is meant to explain, within a fraction of a second. For kinetic theory to correctly predict a gas's pressure and temperature as stable, sustained quantities (which they observably are, for gas sealed in a container), molecular collisions must be essentially perfectly elastic - conserving kinetic energy exactly, collision after collision, indefinitely. ANSWER: Kinetic theory requires molecular collisions to be perfectly elastic because any real energy loss per collision, however small, would compound over the enormous number of collisions happening every second and rapidly bring the gas's molecules to rest - which observably does not happen. To many 18th- and early-19th-century scientists, the idea that countless molecules could keep colliding forever without ever losing any energy at all seemed genuinely implausible, especially before conservation of energy itself was firmly established as a real physical law - which is exactly why Bernoulli's proposal faced real, substantial skepticism for over a century. ---- WHY THIS WORKS AS AN ANSWER This directly connects two separate chapters' own material: Chapter 4's precise distinction between elastic and inelastic collisions explains exactly what property molecular collisions needed to have, and Chapter 3's own history of conservation of energy (not firmly established until the 1840s, a full century after Bernoulli's 1738 proposal) explains why contemporaries lacked the conceptual framework to find "energy-conserving collisions, repeated forever" a natural or credible idea at the time Bernoulli first suggested it.