Question

Air is filled in a motor tube at \(27^{\circ} \mathrm{C}\) and at a Pressure of 8 atmosphere. The tube suddenly bursts. Then what is the temperature of air. given \(\gamma\) of air \(=1.5\) (A) \(150 \mathrm{~K}\) (B) \(150^{\circ} \mathrm{C}\) (C) \(75 \mathrm{~K}\) (D) \(27.5^{\circ} \mathrm{C}\)

Short answer

The temperature of the air after the motor tube bursts is approximately \(150 \mathrm{K}\) (Option A).

Step-by-step solution

Convert the given values to appropriate units

We are given the initial temperature as \(27^{\circ}\mathrm{C}\). To convert it into Kelvin, we use the relation, \(T_1(kelvin) = T_1(celsius) + 273\). Also, we are given that the initial pressure is 8 atmospheres. The SI unit of pressure is Pascal. To convert it to Pascal, we can use the following relation: \[1 \,atm = 101325 \,Pa\]

Calculate the initial temperature in Kelvin

Convert the given temperature (in Celsius) to Kelvin: \(T_1 = 27^{\circ}\mathrm{C} + 273 = 300 \,K\)

Calculate the initial pressure in Pascals

Convert the given pressure (in atmospheres) to Pascal: \(P_1 = 8\,atm \times 101325\,Pa/atm = 810600\,Pa\)

Set the final pressure

The motor tube bursts, and air pressure outside is considered as 1 atmosphere. \(P_2 = 1 \,atm = 101325\,Pa\)

Calculate the final temperature using the adiabatic process formula

Use the given adiabatic equation to find the final temperature: \(T_2 = T_1 \times \left(\frac{P_1}{P_2}\right)^{(\gamma - 1) / \gamma}\) \(T_2 = 300\,K \times \left(\frac{810600\,Pa}{101325\,Pa}\right)^{(1.5 - 1) / 1.5}\)

Calculate the final temperature and convert it back to Celsius

After solving the above equation, we get: \(T_2 = 150\,K\) Now, convert the final temperature back to Celsius: \(T_2 = 150\,K - 273 = -123^{\circ}\mathrm{C}\) The final temperature after the motor tube bursts is \(-123^{\circ}\mathrm{C}\), which is not among the given options. However, if we closely inspect the options, we notice that the temperature of \(75\,K\) and \(150\,K\) are quite close. So, there must be some approximation in the given options. Therefore, option (A) is the closest and can be considered as the correct answer.