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 0.20416 liters of turpentine spill out of the container.

Step-by-step solution

Convert gallons to liters

First, we need to convert the container's volume from gallons to liters, as we will be using the SI unit for volume in our calculations. The conversion factor is 1 gallon ≈ 3.78541 liters. So, the initial volume of the turpentine in liters is: \(V_i = 5.00 \:\text{gal} \times 3.78541 \:\frac{\text{L}}{\text{gal}} = 18.92705 \:\text{L}\)

Calculate the temperature change

Next, we need to calculate the change in temperature, which is the difference between the final and initial temperatures: \(\Delta T = T_f - T_i = 27.0^{\circ} \mathrm{C} - 15.0^{\circ} \mathrm{C} = 12.0^{\circ} \mathrm{C}\)

Calculate the volume expansion

Now, we can calculate the volume expansion of the turpentine using the volume expansion coefficient, the initial volume, and the change in temperature: \(\Delta V = \beta \times V_i \times \Delta T = (9.00 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}) \times (18.92705 \:\text{L}) \times (12.0^{\circ} \mathrm{C})\) \(\Delta V \approx 0.20416 \:\text{L}\) So, the turpentine expands by approximately 0.20416 liters when the temperature increases.

Calculate the amount of fluid spilled

Finally, since the container was initially filled to the brim, the amount of turpentine that spills out of the container is equal to the volume expansion: Spilled turpentine ≈ \(\Delta V \approx 0.20416 \:\text{L}\) The amount of turpentine that spills out of the container when the temperature increases from \(15.0^{\circ} \mathrm{C}\) to \(27.0^{\circ} \mathrm{C}\) is approximately 0.20416 liters.