Two spherical vessels of equal volume are connected by a narrow tube. The apparatus contains an ideal gas at one atmosphere and 300K. Now if one vessel is immersed in a bath of constant temperature 600K and the other in a bath of constant temperature 300K, then the common pressure will be –
Question
Two spherical vessels of equal volume are connected by a narrow tube. The apparatus contains an ideal gas at one atmosphere and 300K. Now if one vessel is immersed in a bath of constant temperature 600K and the other in a bath of constant temperature 300K, then the common pressure will be –
Solution
The problem can be solved using the ideal gas law, which states that the pressure of a gas is directly proportional to its temperature if the volume and the amount of gas are kept constant.
Step 1: Identify the given information.
- Initial pressure (P1) = 1 atmosphere
- Initial temperature (T1) = 300K
- Final temperature of the first vessel (T2) = 600K
- Final temperature of the second vessel (T2') = 300K
Step 2: Apply the ideal gas law. The ideal gas law can be written 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.
Step 3: Solve for the final pressure. Since the volume and the amount of gas are the same in both vessels, the final pressure (P2) will be the same in both vessels. Therefore, we can write the equation as P1/T1 = P2/T2 = P2'/T2'.
Step 4: Substitute the given values into the equation. 1 atmosphere / 300K = P2 / 600K = P2' / 300K
Step 5: Solve for P2 and P2'. P2 = 2 atmospheres P2' = 1 atmosphere
Step 6: Since the pressure is the same in both vessels, the common pressure will be the average of P2 and P2'. Common pressure = (P2 + P2') / 2 = (2 atmospheres + 1 atmosphere) / 2 = 1.5 atmospheres
Therefore, the common pressure will be 1.5 atmospheres.
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