An aluminum vessel with a volume capacity of 510 cm3 is filled with water to the brim at 20.0°C. The vessel and contents are heated heated to 49.9°C. During the heating process, will the water spill over the top (in this case the answer would be positive), will there be more room for water to be added (the answer would be negative), or will the water level remain the same? Calculate the volume of water that will spill over or that could be added.
To solve this problem, we need to consider the thermal expansion properties of water and aluminum.
Step 1: Calculate the initial volume of water
The initial volume of water is given as the vessel's volume capacity, which is 510 cm³.
Step 2: Calculate the final volume of water
To calculate the final volume, we need to determine the change in volume due to thermal expansion. The formula for thermal expansion is:
ΔV = V₀ * β * ΔT
where:
ΔV is the change in volume,
V₀ is the initial volume,
β is the coefficient of volumetric expansion,
and ΔT is the change in temperature.
For water, the coefficient of volumetric expansion (β) is approximately 0.00021 °C⁻¹.
ΔT = 49.9 °C - 20.0 °C = 29.9 °C
ΔV = 510 cm³ * 0.00021 °C⁻¹ * 29.9 °C
Step 3: Determine the final volume
The final volume is the sum of the initial volume and the change in volume:
Final Volume = Initial Volume + Change in Volume
Final Volume = 510 cm³ + ΔV
Now, let's calculate the change in volume and the final volume:
ΔV = 510 cm³ * 0.00021 °C⁻¹ * 29.9 °C
ΔV ≈ 0.318 cm³
Final Volume = 510 cm³ + 0.318 cm³
Final Volume ≈ 510.318 cm³
Step 4: Determine if the water spills over or if there is more room
To determine if the water spills over or if there is more room, we compare the final volume to the vessel's volume capacity:
If Final Volume > Volume Capacity: The water spills over.
If Final Volume = Volume Capacity: The water level remains the same.
If Final Volume < Volume Capacity: There is more room.
Comparing:
Final Volume ≈ 510.318 cm³
Volume Capacity = 510 cm³
Since the final volume is slightly larger than the volume capacity, we can conclude that there will be more room for water, and the water level will not spill over.
Step 5: Calculate the volume of water that could be added
To calculate the volume of water that could be added, we subtract the final volume from the volume capacity:
Volume of Water that could be added = Volume Capacity - Final Volume
Volume of Water that could be added = 510 cm³ - 510.318 cm³
Volume of Water that could be added ≈ -0.318 cm³
The negative sign indicates that the final volume is slightly larger than the volume capacity, and therefore no additional water can be added.
To determine if the water in the aluminum vessel will spill over, remain the same, or allow for more water to be added, we need to utilize the concept of thermal expansion.
Thermal expansion refers to the increase in volume of a material as its temperature rises. In the case of liquids, including water, the expansion is fairly significant.
To calculate the volume of water that will spill over or that could be added, we need to consider the change in volume between the initial and final temperatures. The formula for volumetric thermal expansion of a liquid is given by:
ΔV = β * V₀ * ΔT
where:
ΔV = change in volume
β = coefficient of volumetric expansion of water (for most liquids, this value is approximately 0.00021/°C)
V₀ = initial volume
ΔT = change in temperature
Let's calculate the change in volume:
ΔV = 0.00021/°C * 510 cm³ * (49.9°C - 20.0°C)
ΔV = 0.00021 * 510 cm³ * 29.9°C
ΔV ≈ 3.812 cm³
Since ΔV is the change in volume, a positive value means the volume has increased. Therefore, the water will spill over the top of the vessel during the heating process.
The volume of water that will spill over is approximately 3.812 cm³.