Published by:
CGP EDU Academic Team
Published on: September 12, 2026
A cylinder closed at both ends is divided into two equal parts by a heat-proof piston. Both parts of the cylinder contain the same masses of gas at a temperature t 0 = 27ºC and a pressure P 0 = 1 atm.
What distance from the middle of the cylinder will the piston be displaced if the gas in one of the parts is heated to t = 57ºC? What will be the pressure in this case in each part of the cylinder? The length of half the cylinder is l = 42 cm.
Text Solution
Verified by ExpertsThe correct answer is:
C
Step 1: Understanding the initial conditions
We have a cylinder divided into two equal parts, each with a gas at an initial temperature of t0 = 27ºC (which is 300 K) and an initial pressure P0 = 1 atm. The length of half the cylinder is l = 42 cm.
Step 2: Converting temperatures
The final temperature of the gas in one part is t = 57ºC, which is 330 K. We need to find the change in pressure and the displacement of the piston.
Step 3: Applying the Ideal Gas Law
According to the Ideal Gas Law, for a fixed volume, the pressure P of a gas is given by:
P = nRT/V
Since we are not changing the amount of gas, the number of moles n and volume V for the gas in the unheated side remains constant while the other side changes.
Step 4: Finding new pressure in the heated part
Let P1 be the pressure of the heated gas after heating to 57ºC (330 K). Assuming the volume of gas remains the same initially (volume V1 = V2), we can set up the relation based on temperature and pressure:
$\frac{P_1}{T_1} = \frac{P_0}{T_0}$
Where:
- P0 = 1 atm (initial pressure)
- T0 = 300 K (initial temperature)
- T1 = 330 K (final temperature)
Substituting these values gives us:
P1 = P0 * (T1/T0) = 1 atm * (330 K / 300 K) = 1.1 atm.
Step 5: Finding pressure in the other part
Since the piston moves to balance the pressure, let P2 be the pressure in the other half before displacement. Initially, P2 = P0 = 1 atm (since it remains untouched). After the piston displaces, the pressures must balance:
P1 = P2' (after displacement). Assuming the piston moves a distance \( x \) from the center, the volumes change and can be related to pressures. We can use the relation
P2' * V2' = P2 * V2 (based on initial conditions) which leads to:
1 atm * (l - x) = 1.1 atm * (x).
Step 6: Ratio of Displacement
Solving the above equation leads to:
1 * (42 cm - x) = 1.1 * x
=> 42 - x = 1.1x
=> 42 = 2.1x
=> x = 42/2.1 (displacement towards the higher pressure side).
Calculating gives x = 20 cm.
Conclusion
The piston will be displaced 20 cm towards the heated side, and the pressures in both parts will be 1.1 atm in the heated side and 0.95 atm in the unheated side.
Thus, the answer is:
The piston is displaced 20 cm, and the pressures change accordingly.
We have a cylinder divided into two equal parts, each with a gas at an initial temperature of t0 = 27ºC (which is 300 K) and an initial pressure P0 = 1 atm. The length of half the cylinder is l = 42 cm.
Step 2: Converting temperatures
The final temperature of the gas in one part is t = 57ºC, which is 330 K. We need to find the change in pressure and the displacement of the piston.
Step 3: Applying the Ideal Gas Law
According to the Ideal Gas Law, for a fixed volume, the pressure P of a gas is given by:
P = nRT/V
Since we are not changing the amount of gas, the number of moles n and volume V for the gas in the unheated side remains constant while the other side changes.
Step 4: Finding new pressure in the heated part
Let P1 be the pressure of the heated gas after heating to 57ºC (330 K). Assuming the volume of gas remains the same initially (volume V1 = V2), we can set up the relation based on temperature and pressure:
$\frac{P_1}{T_1} = \frac{P_0}{T_0}$
Where:
- P0 = 1 atm (initial pressure)
- T0 = 300 K (initial temperature)
- T1 = 330 K (final temperature)
Substituting these values gives us:
P1 = P0 * (T1/T0) = 1 atm * (330 K / 300 K) = 1.1 atm.
Step 5: Finding pressure in the other part
Since the piston moves to balance the pressure, let P2 be the pressure in the other half before displacement. Initially, P2 = P0 = 1 atm (since it remains untouched). After the piston displaces, the pressures must balance:
P1 = P2' (after displacement). Assuming the piston moves a distance \( x \) from the center, the volumes change and can be related to pressures. We can use the relation
P2' * V2' = P2 * V2 (based on initial conditions) which leads to:
1 atm * (l - x) = 1.1 atm * (x).
Step 6: Ratio of Displacement
Solving the above equation leads to:
1 * (42 cm - x) = 1.1 * x
=> 42 - x = 1.1x
=> 42 = 2.1x
=> x = 42/2.1 (displacement towards the higher pressure side).
Calculating gives x = 20 cm.
Conclusion
The piston will be displaced 20 cm towards the heated side, and the pressures in both parts will be 1.1 atm in the heated side and 0.95 atm in the unheated side.
Thus, the answer is:
The piston is displaced 20 cm, and the pressures change accordingly.
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