Question

On a cool morning, with the temperature at \(15.0^{\circ} \mathrm{C}\), a painter fills a 5.00 -gal aluminum container to the brim with turpentine. When the temperature reaches \(27.0^{\circ} \mathrm{C}\), how much fluid spills out of the container? The volume expansion coefficient for this brand of turpentine is \(9.00 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\).

Short answer

Answer: Approximately 1.88 liters of fluid spills out of the container.

Step-by-step solution

Calculate the change in temperature

To find the change in temperature, subtract the initial temperature from the final temperature: ΔT = T_final - T_initial ΔT = 27.0°C - 15.0°C ΔT = 12.0°C

Convert volume from gallons to liters

We need to convert the volume of the container from gallons to liters since we will be using the SI unit system. Conversion factor: 1 gallon = 3.78541 liters Initial volume of the container (V_container_initial) = 5.00 gallons V_container_initial = 5.00 * 3.78541 V_container_initial ≈ 18.927 liters

Calculate the volume expansion of turpentine

We'll now use the formula for volume expansion of the turpentine to determine how much it expands when the temperature increases. The formula for volume expansion is: ΔV_turpentine = V_turpentine_initial * β * ΔT Here, β is the volume expansion coefficient given as \(9.00 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\). First, let's find the initial volume of the turpentine (V_turpentine_initial), which is equal to the initial volume of the container. V_turpentine_initial = V_container_initial ≈ 18.927 liters Now we can calculate the volume expansion of turpentine: ΔV_turpentine = 18.927 * \(9.00 \cdot 10^{-4}\) * 12.0 ΔV_turpentine ≈ 2.04 liters

Calculate the volume expansion of the container

Using the formula for volume expansion, we'll now calculate the volume expansion of the container: ΔV_container = V_container_initial * α * ΔT α is the volume expansion coefficient for aluminum which is equal to \(7.2 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1}\). ΔV_container = 18.927 * \(7.2 \cdot 10^{-5}\) * 12.0 ΔV_container ≈ 0.16 liters

Calculate how much fluid spills out of the container

As both the turpentine and the container expand, the difference between their volume expansions will give us the amount of fluid spilled. Spilled fluid = ΔV_turpentine - ΔV_container Spilled fluid ≈ 2.04 liters - 0.16 liters Spilled fluid ≈ 1.88 liters Therefore, when the temperature reaches 27.0°C, approximately 1.88 liters of fluid spills out of the container.