An industrial gas cylinder is equipped with a pressure safety relief valve calibrated to o… — JAMB UTME

An industrial gas cylinder is equipped with a pressure safety relief valve calibrated to open when the internal gas pressure reaches $3.60\text{ atm}$ . At an ambient temperature of $27^\circ\text{C}$ , the enclosed gas exerts a pressure of $2.40\text{ atm}$ . Assuming the volume of the container remains strictly constant, what is the maximum temperature in degrees Celsius ( $^\circ\text{C}$ ) to which the cylinder can be heated before the safety valve opens?

A
$40.5^\circ\text{C}$
$177^\circ\text{C}$ Answer
C
$200^\circ\text{C}$
D
$450^\circ\text{C}$

Answer

The maximum temperature before the safety valve opens is

177^\circ\text{C}

According to the Pressure Law, for a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature (

\frac{P_1}{T_1} = \frac{P_2}{T_2}

). Converting

27^\circ\text{C}

to absolute temperature yields

T_1 = 300\text{ K}

. Substituting

P_1 = 2.40\text{ atm}

P_2 = 3.60\text{ atm}

, and

T_1 = 300\text{ K}

gives

T_2 = 450\text{ K}

. Converting

450\text{ K}

back to Celsius gives

450 - 273 = 177^\circ\text{C}

Step-by-Step Solution

Convert the initial temperature from Celsius to Kelvin

T_1 = 27^\circ\text{C} + 273 = 300\text{ K}

Gas laws require thermodynamic temperature expressed in absolute units (Kelvin).

Apply the Pressure Law (Gay-Lussac's Law) formula at constant volume

\frac{P_1}{T_1} = \frac{P_2}{T_2} \implies \frac{2.40}{300} = \frac{3.60}{T_2}

For a fixed mass of gas at constant volume, pressure is directly proportional to absolute temperature.

Solve for the final absolute temperature

T_2

T_2 = 300 \times \frac{3.60}{2.40} = 450\text{ K}

Cross-multiplying gives the threshold temperature in Kelvin.

Convert the absolute temperature back to degrees Celsius

t_2 = 450\text{ K} - 273 = 177^\circ\text{C}

The question specifically requests the maximum temperature in degrees Celsius.

Key Concept

Pressure Law (Gay-Lussac's Law of Temperature-Pressure)

Estimated Time:1m 30s

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