A 1-liter aluminum container at 20o
C is filled with 0.975 liters of mercury at
20o
C. If both the container and the mercury are heated, at what final temperature
will the mercury completely fill the container? (Hint: this is a volume expansion
problem)
αAl = 24 × 10-6
(Cº
)
-1
; βHg = 1.82×10-4
(°C)-1
changevolumeAl=changeVolHg+.025liters
1*coeffvolAl(tf-20)=.975coeffHg*(tf-20)+.025
coefficent of volume expansion for Aluminum is 3*linear expansion coeff
257
To find the final temperature at which the mercury completely fills the container, we need to consider the expansion of both the aluminum container and the mercury. The volume expansion of the container and the volume expansion of the mercury must be equal for the mercury to completely fill the container.
Let's label the initial volume of the container as V1 and the initial volume of the mercury as V2.
Given:
Initial volume of the container (V1) = 1 liter
Initial volume of the mercury (V2) = 0.975 liters
The expansion of the container can be calculated using the formula:
ΔV1 = V1 * αAl * ΔT
where ΔV1 is the change in volume of the container, αAl is the coefficient of linear expansion of aluminum, and ΔT is the change in temperature.
Similarly, the expansion of the mercury can be calculated using the formula:
ΔV2 = V2 * βHg * ΔT
where ΔV2 is the change in volume of the mercury, βHg is the coefficient of volume expansion of mercury, and ΔT is the change in temperature.
Since both the expansions must be equal for the mercury to completely fill the container, we can set up the equation:
V1 + ΔV1 = V2 + ΔV2
Substituting the values, we get:
1 + V1 * αAl * ΔT = 0.975 + V2 * βHg * ΔT
Simplifying the equation:
1 + 1 * 24 × 10^-6 * ΔT = 0.975 + 0.975 * 1.82 × 10^-4 * ΔT
Now, solve this equation for ΔT to find the change in temperature, which will give us the final temperature at which the mercury completely fills the container.