A 3.00-liter aluminum cylinder at 5.00°C is filled to the brim with gasoline at the same temperature. If the aluminum and gasoline are warmed to 52.0°C, how much gasoline spills out? [Hint: Be sure to account for the expansion of the container. Also, ignore the possibility of evaporation, and assume the volume coefficients are good to three digits. where the average coefficient of linear expansion of aluminum is 24x10^-6 degrees C^-1 and the average coefficient of volume expansion of gasoline is 9.6x10^-4 degrees C^-1
To solve this problem, we need to consider the expansion of both the aluminum cylinder and the gasoline as they are heated.
Step 1: Calculate the change in volume of the aluminum cylinder.
The change in volume of the aluminum cylinder can be calculated using its coefficient of linear expansion and the temperature change. The equation for change in volume (ΔV_aluminum) is given by:
ΔV_aluminum = V_initial * α_aluminum * ΔT
where:
V_initial is the initial volume of the aluminum cylinder (3.00 liters),
α_aluminum is the coefficient of linear expansion of aluminum (24x10^-6 degrees C^-1),
ΔT is the change in temperature (52.0°C - 5.00°C).
ΔV_aluminum = 3.00 liters * (24x10^-6 degrees C^-1) * (52.0°C - 5.00°C)
ΔV_aluminum = 0.003 liters
Step 2: Calculate the change in volume of gasoline.
The change in volume of the gasoline can be calculated using its coefficient of volume expansion and the temperature change. The equation for change in volume (ΔV_gasoline) is given by:
ΔV_gasoline = V_initial * β_gasoline * ΔT
where:
V_initial is the initial volume of the gasoline (3.00 liters),
β_gasoline is the coefficient of volume expansion of gasoline (9.6x10^-4 degrees C^-1),
ΔT is the change in temperature (52.0°C - 5.00°C).
ΔV_gasoline = 3.00 liters * (9.6x10^-4 degrees C^-1) * (52.0°C - 5.00°C)
ΔV_gasoline = 0.1669 liters
Step 3: Calculate the total change in volume.
The total change in volume is the sum of the changes in volume of the aluminum cylinder and the gasoline.
Total change in volume (ΔV_total) = ΔV_aluminum + ΔV_gasoline
ΔV_total = 0.003 liters + 0.1669 liters
ΔV_total = 0.1699 liters
Step 4: Find how much gasoline spills out.
If the container is filled to the brim before heating, then the amount of spilled gasoline is equal to the total change in volume.
Amount of gasoline spilled = ΔV_total
Amount of gasoline spilled = 0.1699 liters
Therefore, approximately 0.1699 liters (or 169.9 mL) of gasoline spills out.
To find out how much gasoline spills out when the temperature increases, we need to calculate the change in volume for both the aluminum cylinder and the gasoline.
Let's start with the aluminum cylinder:
1. Calculate the change in temperature: ΔT = T2 - T1 = 52.0°C - 5.00°C = 47.0°C
(ΔT represents the change in temperature, T2 is the final temperature, and T1 is the initial temperature.)
2. Calculate the change in volume for the aluminum cylinder:
ΔV_aluminum = V_aluminum * α_aluminum * ΔT
(ΔV_aluminum represents the change in volume of the aluminum cylinder, V_aluminum is the initial volume of the aluminum cylinder, α_aluminum is the coefficient of linear expansion of aluminum, and ΔT is the change in temperature.)
ΔV_aluminum = 3.00 L * 24x10^-6 1/°C * 47.0°C
(Using the given values of the initial volume and the coefficient of linear expansion of aluminum.)
ΔV_aluminum = 3.00 L * 0.000024 * 47.0
(Converting the coefficient of linear expansion to decimal form.)
ΔV_aluminum ≈ 0.004248 L ≈ 4.248 mL
(Approximating the result to three decimal places.)
Now, let's calculate the change in volume for the gasoline:
1. Calculate the change in volume for the gasoline:
ΔV_gasoline = V_gasoline * β_gasoline * ΔT
(ΔV_gasoline represents the change in volume of the gasoline, V_gasoline is the initial volume of the gasoline, β_gasoline is the coefficient of volume expansion of gasoline, and ΔT is the change in temperature.)
ΔV_gasoline = 3.00 L * 9.6x10^-4 1/°C * 47.0°C
(Using the given values of the initial volume and the coefficient of volume expansion of gasoline.)
ΔV_gasoline = 3.00 L * 0.00096 * 47.0
(Converting the coefficient of volume expansion to decimal form.)
ΔV_gasoline ≈ 0.12672 L ≈ 126.72 mL
(Approximating the result to three decimal places.)
Since both the aluminum cylinder and the gasoline expand when heated, the total change in volume is the sum of the changes in volume for both:
Total change in volume = ΔV_aluminum + ΔV_gasoline
Total change in volume ≈ 4.248 mL + 126.72 mL
Total change in volume ≈ 130.968 mL
Therefore, approximately 130.968 mL of gasoline spills out when the aluminum cylinder and the gasoline are warmed to 52.0°C.
new volume of the gasonline= (9.6E-4*47 + 1)3=3.14liters
new volume of aluminum cylinder:
if one assumes the change in radius is proportional to change in height, then
newV=V*(1+3*24E-6*47)3=3.01 1liters
so the difference in the change in volumes is ....