Consider a cube of an unknown material. When heated by 10 K, each side expands by 1%. Using *only this information*, what is the ratio between the coefficent of volume expansion and the coefficient of linear expansion (i.e., beta/alpha)? Express your answer to two significant figures.
3.0 is always the ratio.
The temperature rise and linear expansion amount don't matter.
To find the ratio between the coefficient of volume expansion and the coefficient of linear expansion, we need to understand the relationship between them.
The coefficient of linear expansion (α) measures how much a material expands linearly per unit change in temperature. It is defined as the fractional change in length of a material per unit change in temperature.
The coefficient of volume expansion (β), on the other hand, quantifies how much a material expands in volume per unit change in temperature. It is defined as the fractional change in volume of a material per unit change in temperature.
In order to find the ratio β/α, we need to relate these two coefficients. One way to do this is by considering the expansion of a cube.
Let's consider a cube with an edge length of 1 unit, made of the unknown material. When the temperature increases by 10 K, each side expands by 1% or 0.01 units (1% of 1 unit).
The original volume of the cube is V = (1 unit)^3 = 1 unit^3.
After expansion, each side will have a length of 1 unit + 0.01 units = 1.01 units.
The new volume of the expanded cube is V' = (1.01 units)^3 = 1.030301 unit^3.
The change in volume (ΔV) is given by ΔV = V' - V = 1.030301 unit^3 - 1 unit^3 = 0.030301 unit^3.
Now, the fractional change in volume is ΔV/V = 0.030301 unit^3 / 1 unit^3 = 0.030301.
We also know that the fractional change in length is given by ΔL/L, where ΔL is the change in length (0.01 units) and L is the original length (1 unit).
Therefore, ΔL/L = 0.01/1 = 0.01.
Since ΔV/V = βΔT and ΔL/L = αΔT (where ΔT is the change in temperature), we can equate these two expressions:
βΔT = αΔT.
We can cancel out ΔT to get β = α.
Therefore, the ratio β/α is 1.
Thus, the ratio between the coefficient of volume expansion and the coefficient of linear expansion is 1 (to two significant figures).