Question

Consider two geometrically identical cylinders of inner diameter \(5.00 \mathrm{~cm},\) one made of copper and the other of Teflon, each closed on the bottom and open on top. The two cylinders are immersed in a large-volume water tank at room temperature \(\left(20.0^{\circ} \mathrm{C}\right)\), as shown in the figure. Note that the copper cylinder is an excellent conductor of heat and the Teflon cylinder is a good insulator. A frictionless piston with a rod and platter attached is placed in each cylinder. The mass of the piston-rod- platter assembly is \(0.500 \mathrm{~kg}\), and the cylinders are filled with helium gas so that initially both pistons are at equilibrium at \(20.0 \mathrm{~cm}\) from the bottom of their respective cylinders a) A 5.00 -kg lead block is slowly placed on each platter, and the piston is slowly lowered until it reaches its final equilibrium state. Calculate the final height of each piston (as measured from the bottom of its cylinder). b) If the two lead blocks are dropped suddenly on the platters, how will the final heights of the two pistons compare immediately after the lead blocks are dropped?

Short answer

Answer: When the 5 kg lead blocks are added, the final height of the pistons in both cylinders will decrease due to the increased pressure. In the copper cylinder, the process is isothermal due to its good thermal conductivity, whereas in the Teflon cylinder, the process is adiabatic due to its insulating properties. If the blocks were dropped suddenly, the final height in the copper cylinder would remain the same, while the final height in the Teflon cylinder would be slightly lower due to the rapid change in pressure and temperature.

Step-by-step solution

Calculate the initial pressure

First, calculate the initial pressure of the helium gas when the pistons are at equilibrium using the formula: P = (m * g) / A where m = 0.5 kg (mass of piston-rod-platter assembly), g = 9.81 m/s^2 (gravitational acceleration), and A = π * (d / 2)^2 (the cross-sectional area of the cylinder with a diameter d = 0.05 m).

Calculate the initial volume of the gas in both cylinders

The initial volume of the gas in both cylinders can be calculated using the formula: V_initial = A * h where h = 0.2 m (initial height of the piston from the bottom).

Calculate the final pressure in both cylinders

Add the pressure due to the 5 kg lead block to the initial pressure to find the final pressure in both cylinders: P_final = P_initial + (M_block * g) / A where M_block = 5 kg (mass of the lead block).

Calculate the final height of the piston in the copper cylinder

Since the copper cylinder is an excellent conductor of heat, we can assume this process to be isothermal, meaning the temperature remains constant. Therefore, for the copper cylinder, we can use the ideal gas law in the form of Boyle's law, which states that for an isothermal process, P_initial * V_initial = P_final * V_final. So, solving for the final height of the piston in the copper cylinder: V_final_copper = V_initial * (P_initial / P_final) h_final_copper = V_final_copper / A

Calculate the final height of the piston in the Teflon cylinder

For the Teflon cylinder, it is a good insulator, so we can assume this process to be adiabatic, meaning no heat is exchanged. For an adiabatic process, we can use the relationship: P_initial * V_initial^γ = P_final * V_final^γ where γ = 5/3 (the adiabatic index for a monoatomic gas like helium). Solving for the final height of the piston in the Teflon cylinder: V_final_teflon = V_initial * (P_initial / P_final)^(1 / γ) h_final_teflon = V_final_teflon / A b) Compare the final heights of the pistons when the lead blocks are dropped suddenly:

Consider the effects of sudden pressure change

When the lead blocks are dropped suddenly, the pressure inside the cylinders changes instantaneously. Due to the difference in thermal conductivity, the copper and Teflon cylinders will respond differently.

Compare the final heights

For the copper cylinder, since it is a good conductor, it can quickly equalize the heat generated due to the sudden pressure increase, and the final height will be the same as in part a (isothermal process). For the Teflon cylinder, since it is a good insulator, if the lead blocks are dropped suddenly, the process is closer to an adiabatic process. The Teflon cylinder will experience a greater temperature increase compared to the copper cylinder, leading to a higher pressure inside the cylinder. Therefore, in this case, the final height of the piston in the Teflon cylinder will be slightly lower than that in part a.