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Content Me panic-reviewing gas law calculations at 2 AM for my 7 AM exam. Because n is constant, we can use Equation 10.8. Solve: Rearranging Equation 10.8 to solve for V2 gives ½ = 4 x - (6.0 L) 1.0 atm /252 K 295 K, = 11L 0.45 atm/ check: The result appears reasonable. Notice that the felt temperatures moles, fits the initial voltaebya ratio of pressures endle volume connect sim, the expect that alecreasing pressure will cause the yetuense. increase Sintany, we expect that decre sion id cause the volume to decrease afore st at the dister. in pressures is raote aramatic than the difference in temperateres Thus, we shag expect the effect of the pressure change to predominate in determining the final yo. ume, as it does. PRACTICE EXERCISE A 0.50-mol sample of oxygen gas is confined at 0 °C in a cylinder with a morade piston, such as that shown in Figure 10.12. The gas has an initial pressure of 10 at. The piston then compresses the gas so that its final volume is halt the initial volume The final pressure of the gas is 2.2 atm. What is the final temperature of the gas in degrees Celsius? 10.5 FURTHER APPLICI OF THE IDEAL-GAS EQUATION The ideal-gas equation can be used to determine many relationships involving the physical properties of gases. In this section we use it first to define the rela tionship between the density of a gas and its molar mass, and then to calculate the volumes of gases formed or consumed in chemical reactions Gas Densities and Molar Mass The ideal-gas equation allows us to calculate gas density from the molar mas pressure, and temperature of the gas. Recall that density has the units of me per unit volume (d = m/V). a (Section 1.4) We can arrange the gas equat to obtain similar units, moles per unit volume, n/V: P V RT If we multiply both sides of this equation//// @ sergM,

When you're so deep in mathematical obsession that you start marketing your unsolved millennium problem like it's the next Marvel movie franchise. Nothing says "I've lost touch with reality" quite like releasing a proof in episodic installments while monitoring prediction markets for reactions. The Navier-Stokes equations have claimed another victim! That abstract is what happens when you let ChatGPT write your mathematical papers after feeding it nothing but fluid dynamics textbooks and energy drinks. The real twist ending? The proof was inside us all along... or maybe just inside this guy's imagination.

From "I'm interested in the Collatz conjecture" to emailing a UCLA math professor claiming you've solved it after ChatGPT inflated your ego? That's not a proof, that's a mathematical tragedy in four acts! The Collatz conjecture has stumped brilliant minds for 85+ years, but sure, you "see the pattern" without advanced math. Next you'll be explaining how you've unified quantum mechanics and general relativity while waiting for your coffee to brew. Pro tip: If your mathematical breakthrough involves a rainbow clown wig, perhaps reconsider your life choices.

The mathematical equivalent of seeing Jesus in your toast! This guy's claiming to have solved the Navier-Stokes equations—one of math's million-dollar Millennium Prize Problems—while casually tweeting about it like he's sharing a breakfast recipe. The Navier-Stokes smoothness problem has stumped mathematicians for decades, but apparently all it needed was some "pancake control" and relationship advice. Next up: solving quantum gravity with a TikTok dance! What makes this extra hilarious is the perfect blend of genuine mathematical notation with completely unhinged conclusions. It's the academic equivalent of that 3AM eureka moment when you think you've discovered time travel but actually just need a sandwich and sleep.