Final answer:
To increase the pressure of a gas cylinder to 1.5 times its initial pressure at room temperature, the temperature must be raised to 166.575°C, using the direct relationship between pressure and temperature defined by Gay-Lussac's law.
Step-by-step explanation:
The temperature a cylinder of gas must be increased to in order for the pressure to be 1.5 times the initial pressure p1 can be determined using the Gay-Lussac's law. This law states that the pressure of a gas is directly proportional to the temperature when the volume is constant. Mathematically, it is expressed as P1/T1 = P2/T2, where P1 and T1 are the initial pressure and temperature, and P2 and T2 are the final pressure and temperature. If we're starting with pressure p1 at room temperature, which is approximately 20°C or 293.15 K, and we want to increase the pressure to 1.5p1, we can solve for T2:
P1/T1 = P2/T2
p1/(293.15 K) = 1.5p1/T2
T2 = 1.5(293.15 K)
T2 = 439.725 K
Converting the temperature back to Celsius:
T2 = 439.725 K - 273.15
= 166.575°C
Therefore, the temperature must be increased to 166.575°C for the pressure to become 1.5p1.