A hollow aluminum cylinder 22.0 cm deep has an internal capacity of 2.000 L at 21.0°C. It is completely filled with turpentine at 21.0°C. The turpentine and the aluminium cylinder are then slowly warmed together to 87.0°C. (The average linear expansion coefficient for aluminium is 2.4 10-5°C−1, and the average volume expansion coefficient for turpentine is 9.0 10-4°C−1.)
(a) How much turpentine overflows?
(b) What is the volume of turpentine remaining in the cylinder at 87.0°C? (Give your answer to four significant figures.)
(c) If the combination with this amount of turpentine is then cooled back to 21.0°C, how far below the cylinder's rim does the turpentine's surface recede?
To solve this question, we need to consider the expansion of both the aluminum cylinder and the turpentine as they are heated. We'll use the expansion coefficients provided to calculate the change in volume for each.
(a) To find out how much turpentine overflows, we need to calculate the increase in volume of the turpentine and compare it to the internal capacity of the cylinder.
First, let's calculate the change in volume for the aluminum cylinder:
ΔV_aluminum = V_aluminum * α_aluminum * ΔT
where V_aluminum is the initial volume of the cylinder, α_aluminum is the linear expansion coefficient of aluminum, and ΔT is the change in temperature. Note that the change in volume of the aluminum cylinder does not contribute to overflow since it is hollow.
Next, let's calculate the change in volume for the turpentine:
ΔV_turpentine = V_turpentine * β_turpentine * ΔT
where V_turpentine is the initial volume of the turpentine, β_turpentine is the volume expansion coefficient of turpentine, and ΔT is the change in temperature.
Now, the turpentine overflows if the change in volume ΔV_turpentine is greater than the internal capacity of the cylinder. If ΔV_turpentine exceeds the internal capacity, the excess volume will overflow.
To compute the overflow volume (V_overflow):
V_overflow = ΔV_turpentine - V_cylinder_internal
(b) To calculate the volume of turpentine remaining in the cylinder at 87.0°C, we need to subtract the overflow volume (V_overflow) from the initial volume of the turpentine:
V_remaining = V_turpentine - V_overflow
(c) Finally, to determine how far below the cylinder's rim the turpentine's surface recedes when cooled back to 21.0°C, we need to calculate the change in volume of the turpentine when going from 87.0°C to 21.0°C using the volume expansion coefficient β_turpentine:
ΔV_cooled = V_remaining * β_turpentine * ΔT
where V_remaining is the volume of turpentine remaining in the cylinder at 87.0°C, β_turpentine is the volume expansion coefficient of turpentine, and ΔT is the change in temperature.
Let's plug in the values and calculate the results.