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 problem, we can follow the steps below:
Step 1: Calculate the initial volume of the aluminum cylinder.
Given that the internal capacity of the cylinder is 2.000 L at 21.0°C, we can convert it to cm^3 by multiplying by 1000:
Initial volume = 2.000 L * 1000 cm^3/L = 2000 cm^3
Step 2: Find the change in volume of the aluminum cylinder as it is heated from 21.0°C to 87.0°C.
The change in volume of an object due to temperature change can be calculated using the formula:
∆V_1 = V_1 * α * ∆T
Where:
∆V_1 is the change in volume of the aluminum cylinder.
V_1 is the initial volume of the aluminum cylinder.
α is the linear expansion coefficient for aluminum.
∆T is the change in temperature.
∆V_1 = 2000 cm^3 * (2.4 * 10^-5 °C^-1) * (87.0°C - 21.0°C)
Step 3: Calculate the final volume of the aluminum cylinder after heating.
The final volume can be found by adding the change in volume (∆V_1) to the initial volume (2000 cm^3).
Final volume = Initial volume + ∆V_1
Step 4: Find the change in volume of turpentine.
The change in volume of a liquid due to temperature change can be calculated using the formula:
∆V_2 = V_2 * β * ∆T
Where:
∆V_2 is the change in volume of turpentine.
V_2 is the initial volume of turpentine.
β is the volume expansion coefficient for turpentine.
∆T is the change in temperature.
Since the aluminum cylinder is completely filled with turpentine, the initial volume of turpentine is equal to the initial volume of the aluminum cylinder, which is 2000 cm^3.
∆V_2 = 2000 cm^3 * (9.0 * 10^-4 °C^-1) * (87.0°C - 21.0°C)
Step 5: Calculate the volume of turpentine remaining in the cylinder at 87.0°C.
The volume of turpentine remaining can be found by subtracting the change in volume of turpentine (∆V_2) from the initial volume of turpentine.
Volume of turpentine remaining = Initial volume of turpentine - ∆V_2
Step 6: Calculate the amount of turpentine that overflows.
The amount of turpentine that overflows can be found by subtracting the final volume of the aluminum cylinder from the volume of turpentine remaining.
Overflow = Volume of turpentine remaining - Final volume
Step 7: Calculate how far below the cylinder's rim the turpentine's surface recedes when cooled back to 21.0°C.
The change in volume of turpentine when cooled back can be calculated using the same formula as in Step 4:
∆V_3 = V_3 * β * ∆T
Where:
∆V_3 is the change in volume of turpentine when cooled back.
V_3 is the volume of turpentine remaining at 87.0°C.
β is the volume expansion coefficient for turpentine.
∆T is the change in temperature.
The final volume of turpentine when cooled back to 21.0°C can be found by subtracting the change in volume of turpentine (∆V_3) from the volume of turpentine remaining at 87.0°C.
Now let's plug in the values and calculate the answers:
Step 1: Initial volume of the aluminum cylinder = 2000 cm^3
Step 2: ∆V_1 = 2000 cm^3 * (2.4 * 10^-5 °C^-1) * (87.0°C - 21.0°C)
Step 3: Final volume of the aluminum cylinder = 2000 cm^3 + ∆V_1
Step 4: ∆V_2 = 2000 cm^3 * (9.0 * 10^-4 °C^-1) * (87.0°C - 21.0°C)
Step 5: Volume of turpentine remaining = 2000 cm^3 - ∆V_2
Step 6: Overflow = Volume of turpentine remaining - Final volume
Step 7: ∆V_3 = (Volume of turpentine remaining at 87.0°C) * (9.0 * 10^-4 °C^-1) * (87.0°C - 21.0°C)
Turpentine's surface recedes = (Volume of turpentine remaining at 87.0°C - ∆V_3) / (Area of the cylinder's base)
Please note that in Step 7, we would need the diameter or radius of the cylinder to calculate the area of the base and find how far the turpentine's surface recedes. The question does not provide the dimensions of the cylinder, so we are unable to determine this value.
To solve this problem, we need to consider the expansion of both the aluminum cylinder and the turpentine as they are heated. We can break down the problem into three parts:
Part (a): Calculating the amount of turpentine that overflows.
Part (b): Calculating the volume of turpentine remaining in the cylinder at 87.0°C.
Part (c): Determining how far below the cylinder's rim the turpentine's surface recedes when cooled back to 21.0°C.
Let's solve each part step-by-step.
Part (a): Calculating the amount of turpentine that overflows.
To determine the amount of turpentine that overflows, we need to compare the initial volume of the cylinder to the final volume after expansion.
Given:
Initial internal capacity of the cylinder = 2.000 L
Initial temperature = 21.0°C
Final temperature = 87.0°C
To calculate the final volume of the cylinder, we need to consider the expansion of the aluminum cylinder using its linear expansion coefficient.
The change in temperature = Final temperature - Initial temperature
= 87.0°C - 21.0°C
= 66.0°C
The change in length of the aluminum cylinder can be calculated using its linear expansion coefficient (2.4 x 10^-5 °C^-1) and the initial depth of the cylinder (22.0 cm).
Change in length of the aluminum cylinder = Linear expansion coefficient * initial depth * change in temperature
= 2.4 x 10^-5 °C^-1 * 22.0 cm * 66.0°C
≈ 0.03 cm
The increase in volume of the cylinder due to the expansion is given by:
Increase in volume = π * (radius)^2 * change in length
= π * (radius)^2 * 0.03 cm
Now, we need to convert the increase in volume to liters by dividing by 1000 (since 1 liter = 1000 cm^3).
Increase in volume (in liters) = π * (radius)^2 * 0.03 cm / 1000
To calculate the amount of turpentine that overflows, we subtract the increase in volume from the initial capacity of the cylinder:
Amount of turpentine that overflows = Initial internal capacity - Increase in volume (in liters)
Part (b): Calculating the volume of turpentine remaining in the cylinder at 87.0°C.
The volume of turpentine remaining in the cylinder at 87.0°C is the initial internal capacity minus the amount of turpentine that overflows.
Volume of turpentine remaining (in liters) = Initial internal capacity - Amount of turpentine that overflows
Part (c): Calculating how far below the cylinder's rim the turpentine's surface recedes when cooled back to 21.0°C.
To determine how far below the cylinder's rim the turpentine's surface recedes, we consider the volume contraction of the turpentine as it cools down.
Given:
Final temperature (after cooling) = 21.0°C
To calculate the decrease in volume of the turpentine, we multiply the initial internal capacity by the volume expansion coefficient of turpentine (9.0 x 10^-4 °C^-1) and the change in temperature.
The change in temperature = Final temperature - Initial temperature
= 21.0°C - 87.0°C
= -66.0°C (Note: Negative because we're cooling down)
Decrease in volume of the turpentine = Initial internal capacity * volume expansion coefficient * change in temperature
To convert the decrease in volume to liters, we divide by 1000 (since 1 liter = 1000 cm^3).
Decrease in volume of the turpentine (in liters) = Initial internal capacity * volume expansion coefficient * change in temperature / 1000
Hence, the turpentine's surface recedes below the cylinder's rim by the decrease in volume of the turpentine.
I hope this helps! Let me know if you have any further questions.