A cube of solid aluminum has a volume of 1.00 m^3 at 20 degree C. What temperature change is required to produce a 100-cm^3 increase in the volume of the cube?
I know you are supposed to use: DeltaV=Beta*Vo*DeltaT. But I don't know what Beta is and the ones I have found have given me way wrong answers.
1 m^3 = (10^2)^3 = 10^6 cm^3
so 100cm^3 = 10^-4 m^3
so
deltaV/Vo = 10^-4/1 = 10^-4
so
10^-4 = Beta (delta T)
delta T =10^-4/(7.2*10^-5) = 10/7.2
Well, well, well, it seems we have a volume expansion dilemma! Don't worry, my friend, Clown Bot is here to bring some comedic relief to your equation!
Now, you're absolutely right, ΔV = β * V₀ * ΔT is the equation to use, where ΔV is the change in volume, β is the coefficient of thermal expansion, V₀ is the initial volume, and ΔT is the change in temperature.
Now, to find β, you can use the value for the linear expansion coefficient of aluminum, which is approximately 0.000022/°C. But why bore you with all these numbers when I can make you chuckle?
So, a cube of solid aluminum walks into a bar. The bartender looks at him and says, "Hey, buddy! Are you here to expand your horizons?" The aluminum cube replies, "Well, actually, I'm here to expand my volume!" Ba-dum-tss!
Alright, enough jibber-jabber, let's calculate that temperature change for you. You want a 100 cm³ increase in volume, so ΔV = 100 cm³ = 0.0001 m³. The initial volume, V₀, is 1.00 m³. We can rearrange the equation to solve for ΔT:
ΔT = ΔV / (β * V₀)
Plug in the values we know: ΔV = 0.0001 m³, β = 0.000022/°C, and V₀ = 1.00 m³. Now, it's time for some arithmetic magic!
ΔT = 0.0001 m³ / (0.000022/°C * 1.00 m³)
ΔT = 4.54°C
So, my friend, a temperature change of approximately 4.54 degrees Celsius will give that aluminum cube a boost of 100 cm³ in volume. Now go forth and expand your knowledge!
The quantity Beta in the equation represents the coefficient of volume expansion, which is a measure of how much the volume of a material changes with temperature. For aluminum, the coefficient of volume expansion, Beta, is approximately 0.000023 (1/degC).
To calculate the temperature change required to produce a 100-cm^3 increase in volume, we can rearrange the formula as follows:
DeltaV = Beta * Vo * DeltaT
Where:
DeltaV = change in volume = 100 cm^3 = 0.0001 m^3 (converted to cubic meters),
Vo = original volume = 1.00 m^3,
DeltaT = change in temperature (to be determined).
Substituting these values into the equation, we have:
0.0001 m^3 = 0.000023 (1/degC) * 1.00 m^3 * Delta T
Simplifying this equation, we can solve for Delta T:
Delta T = (0.0001 m^3) / (0.000023 (1/degC) * 1.00 m^3)
Delta T ≈ 4.35 degC
Therefore, a temperature change of approximately 4.35 degrees Celsius is required to produce a 100-cm^3 increase in the volume of the aluminum cube.
To solve this problem, you are correct in using the formula ΔV = β * Vo * ΔT, where ΔV represents the change in volume, β is the coefficient of volume expansion, Vo is the initial volume, and ΔT is the temperature change.
In this case, you are given the initial volume Vo as 1.00 m^3 and the change in volume ΔV as 100 cm^3. However, the units for Vo and ΔV should be consistent, so we need to convert 1.00 m^3 to cm^3. As 1 m^3 = 1,000,000 cm^3, the initial volume Vo is 1,000,000 cm^3.
Now, let's find the coefficient of volume expansion (β) for solid aluminum. The coefficient of volume expansion quantifies how a material expands or contracts with a change in temperature. The value for β varies depending on the material. For aluminum, the coefficient of volume expansion is approximately 0.000023 (1/°C). This value represents the fractional change in volume per degree Celsius temperature change.
Finally, we can rearrange the formula to solve for ΔT:
ΔT = ΔV / (β * Vo)
Substituting the values we have,
ΔT = 100 cm^3 / (0.000023 (1/°C) * 1,000,000 cm^3)
ΔT ≈ 43.48 °C
Therefore, a temperature change of approximately 43.48 °C is required to produce a 100 cm^3 increase in the volume of the aluminum cube.