A high-pressure gas cylinder contains 40.0 L of toxic gas at a pressure of 2.00

Question

A high-pressure gas cylinder contains 40.0 L of toxic gas at a pressure of 2.00 ? 107 N/m2 and a temperature of 18.0°C. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperatures (-78.5°C), to reduce the leak rate and pressure so that it can be safely repaired.

(a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change?
  N/m2
(b) What is the final pressure if one-tenth of the gas escapes?
  N/m2
(c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)?

Explanation / Answer

given that :

volume of toxic gas inside a cylinder, V = 40 L

initial pressure, p1 = 2 x 107 N/m2

initial temperature, T1 = 18 0C = (18 + 273) K = 291K

final temperature, T2 = -78.5 0C = (273 - 78.5) K = 194.5 K

(a) Assuming a negligible amount of gas leaks while being cooled and that there is no phase change, then the final pressure in the tank is given as :

using a Gay-lussac law,

p1 / p2 = T1 / T2                        (volume remains constant)

or p2 = p1 x T2 / T1                         { eq. 1 }

p2 = (2 x 107 N/m2) ( 194.5 K) / (291K)

p2 = 1.33 x 107 N/m2

(b) the final pressure if one-tenth of the gas escapes which is given as ::

using a combined gas law,

p1 V1 / T1 = p2 V2 / T2

or p2 = p1 (V1 / V2) (T2 / T1)                   { eq. 2 }

where, V2 / V1 = 1 / 10

(V1 / V2) = 10

inserting the values in eq.2,

p2 = (2 x 107 N/m2) (10) [(194.5 K ) / (291K)]

p2 = 13.36 x 107 N/m2

or p2 = 1.33 x 106 N/m2

(c) when final pressure is 1 atm, then its final temperature is given as ::

using a formula of "Gay-lussac law",

p1 / p2 = T1 / T2                        (volume remains constant)

or T2 = p2 T1 / p1                        { eq. 3 }

where, p2 = 1 atm = 1 x 105 N/m2

inserting the values in eq.3

T2 = (1 x 105 N/m2) (291K) / (2 x 107 N/m2)

T2 = 145.5 x 10-2 K

T2 = 1.45 K

Temperature in kelvin change into degree celsius :

T2 = (1.45 - 273) 0C

T2 = - 271.5 0C

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