Two discharge tubes possess identical material structure and are filled with the same gas. The length of one tube is 10 cm while that of the second tube is 20 cm . Sparking initiates in both tubes when the potential difference between the cathode and the anode reaches 100 V . If the pressure in the shorter tube is 1.0 mm of mercury, determine the pressure in the longer tube.

Step-by-step solution

According to Paschen's law, the sparking potential V of a gas between two electrodes depends only on the product of the gas pressure p and the distance d between the electrodes. V = f(pd) Since the sparking potential is the same for both tubes, the product of pressure and length must be constant. p₁ d₁ = p₂ d₂ Substituting the given values p₁ = 1.0 mm of Hg, d₁ = 10 cm , and d₂ = 20 cm : 1.0 10 = p₂ 20 p₂ = 10 20 = 0.5 mm of Hg