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Air is filled in a bottle and it is cork...

Air is filled in a bottle and it is corked at `35^(@)C`. If the cork can come out at 3 atmospheric pressure, then upto what temperature should the bottle be heated to remove the cork ?

A

`325.5^(@)C`

B

`851^(@)C`

C

`651^(@)C`

D

None of these

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will use the ideal gas law, which states that \( PV = nRT \). Here’s a step-by-step solution: ### Step 1: Understand the Initial Conditions We have a bottle filled with air at an initial temperature of \( 35^\circ C \). We need to convert this temperature to Kelvin for our calculations. \[ T_1 = 35 + 273 = 308 \, K \] ### Step 2: Identify the Initial Pressure The initial pressure \( P_1 \) is given as 1 atmospheric pressure (1 atm). ### Step 3: Set Up the Ideal Gas Law for Initial Conditions Using the ideal gas law, we can express the initial conditions as: \[ P_1 V = n R T_1 \] Substituting the known values: \[ 1 \cdot V = n R \cdot 308 \] ### Step 4: Identify the Final Conditions The cork can come out when the pressure reaches 3 atmospheric pressure. Therefore, the final pressure \( P_2 \) is: \[ P_2 = 3 \, \text{atm} \] ### Step 5: Set Up the Ideal Gas Law for Final Conditions For the final conditions, we can express it as: \[ P_2 V = n R T_2 \] Substituting the known values: \[ 3 \cdot V = n R \cdot T_2 \] ### Step 6: Relate the Two Equations Since the volume \( V \) and the number of moles \( n \) are constant, we can set the two equations equal to each other: \[ 1 \cdot V = n R \cdot 308 \quad \text{(initial)} \] \[ 3 \cdot V = n R \cdot T_2 \quad \text{(final)} \] ### Step 7: Cancel Out Common Terms Dividing the second equation by the first gives us: \[ \frac{3V}{1V} = \frac{nRT_2}{nR \cdot 308} \] This simplifies to: \[ 3 = \frac{T_2}{308} \] ### Step 8: Solve for \( T_2 \) Now, we can solve for \( T_2 \): \[ T_2 = 3 \cdot 308 = 924 \, K \] ### Step 9: Convert \( T_2 \) to Celsius To find the temperature in degrees Celsius, we convert from Kelvin: \[ T_2 = 924 - 273 = 651^\circ C \] ### Final Answer The bottle should be heated to \( 651^\circ C \) to remove the cork. ---
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