An aluminum sphere is 8.60 in diameter. What will be its percent change in volume if it is heated from 30 to 120 degrees Celsius.
To calculate the percent change in volume, we first need to determine the initial and final volumes of the aluminum sphere.
The formula for the volume of a sphere is V = (4/3) * π * r^3, where V represents the volume and r is the radius of the sphere.
Given that the diameter of the aluminum sphere is 8.60 inches, we can calculate its initial radius by dividing the diameter by 2:
Initial radius (r1) = diameter / 2 = 8.60 / 2 = 4.30 inches
We can then substitute the initial radius into the volume formula:
Initial volume (V1) = (4/3) * π * r1^3
Now we need to find the final volume of the sphere when it is heated from 30 to 120 degrees Celsius. The expansion of solids can be calculated using the linear expansion coefficient (α) and the equation ΔL = αL0ΔT, where ΔL is the change in length, α is the linear expansion coefficient, L0 is the initial length, and ΔT is the change in temperature.
The linear expansion coefficient for aluminum is typically around 0.000022 per degree Celsius.
In this case, since we are dealing with a sphere, the change in radius (Δr) can be calculated by αr0ΔT, where Δr is the change in radius, α is the linear expansion coefficient, r0 is the initial radius, and ΔT is the change in temperature.
Change in radius (Δr) = α * r0 * ΔT
Substituting the values, we get:
Change in radius (Δr) = (0.000022) * (4.30 inches) * (120°C - 30°C)
Change in radius (Δr) = 0.000022 * 4.30 * 90 = 0.008214 inches
The final radius (r2) is obtained by adding the change in radius to the initial radius:
Final radius (r2) = r1 + Δr = 4.30 + 0.008214 = 4.308214 inches
Now we can calculate the final volume (V2) of the sphere using the final radius:
Final volume (V2) = (4/3) * π * r2^3
With V1 and V2 determined, we can now find the percent change in volume using the following formula:
Percent change in volume = [(V2 - V1) / V1] * 100
By substituting the calculated values and performing the arithmetic, we can determine the percent change in volume of the aluminum sphere when heated from 30 to 120 degrees Celsius.