In an air-conditioned room at 19.0 ∘C, a spherical balloon had the diameter of 50.0 cm. When taken outside on a hot summer day, the balloon expanded to 51.0 cm in diameter. What was the temperature outside in degrees Celsius? Assume that the balloon is a perfect sphere and that the pressure and number of moles of air molecules remains the same.
A cylinder with a movable piston contains 2.00 g of helium, He, at room temperature. More helium was added to the cylinder and the volume was adjusted so that the gas pressure remained the same. How many grams of helium were added to the cylinder if the volume was changed from 2.00 L to 2.50 L ? (The temperature was held constant.)
+6251 \(PV=nRT\)
\(V_1=\dfrac 4 3 \pi \left(\dfrac {50}{2}\right)^3 \\ V_2 = \dfrac 4 3 \pi \left(\dfrac {51}{2}\right)^3 \\ \mbox{The rest of the quantities other than temperature stay constant.} \)
\(\dfrac{P V_1}{P V_2} = \dfrac{n R T_1}{n R T_2} \Rightarrow \dfrac {V_1}{V_2}=\dfrac {T_1}{T_2}\)
\(\dfrac {V_1}{V_2}=\dfrac{50^3}{51^3} = \dfrac {125000}{132651} \\ \mbox{so }\dfrac {T_1}{T_2} = \dfrac {125000}{132651} \\ T_2 = T_1 \dfrac{132651}{125000} = 19 \dfrac{132651}{125000} = 20.163 ^\circ C\)
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