Two compressible solids are formed into spheres of the same size. The bulk modulus of sphere two is twice as large as the bulk modulus of sphere one. You now increase the pressure on both spheres by the same amount. As a result of the increased pressure, how is the change in volume of sphere two (△V2) related to the change in volume of sphere one (ΔV1)?
A) ΔV2 = 2ΔV1
B) ΔV2 = 4ΔV1
C) ΔV2 = ΔV1
D) ΔV2 = 1/2ΔV1
E) ΔV2 = 1/4ΔV1
B) ΔV2 = 4ΔV1
To understand how the change in volume of sphere two (ΔV2) is related to the change in volume of sphere one (ΔV1), we need to consider the bulk modulus of the two spheres.
The bulk modulus (K) is a measure of how resistant a material is to changes in volume under pressure. A higher bulk modulus indicates a greater resistance to deformation.
Given that the bulk modulus of sphere two is twice as large as the bulk modulus of sphere one, we can denote the bulk modulus of sphere one as K1 and the bulk modulus of sphere two as 2K1 (since it is twice as large).
When pressure is applied to a material, the change in volume is related to the bulk modulus through the equation:
ΔV = -V * ΔP / K
Where ΔV is the change in volume, V is the initial volume, ΔP is the change in pressure, and K is the bulk modulus.
Let's apply this equation to the two spheres. Since we are increasing the pressure on both spheres by the same amount, ΔP is the same for both spheres.
For sphere one:
ΔV1 = -V1 * ΔP / K1
And for sphere two:
ΔV2 = -V2 * ΔP / (2K1)
Now, let's compare the changes in volume ΔV1 and ΔV2:
ΔV2 / ΔV1 = (-V2 * ΔP / (2K1)) / (-V1 * ΔP / K1)
ΔV2 / ΔV1 = -V2 * ΔP * K1 / (-V1 * ΔP * 2K1)
The ΔP and K1 terms cancel out:
ΔV2 / ΔV1 = -V2 / -V1 * 1 / 2
Finally, we simplify the expression:
ΔV2 / ΔV1 = V2 / V1 * 1 / 2
Since the spheres are formed into spheres of the same size, V2 / V1 = 1. Therefore:
ΔV2 / ΔV1 = 1 * 1 / 2
ΔV2 / ΔV1 = 1 / 2
Hence, the change in volume of sphere two (ΔV2) is equal to half the change in volume of sphere one (ΔV1).
Therefore, the correct answer is:
D) ΔV2 = 1/2ΔV1
The relationship between the change in volume of sphere two (ΔV2) and the change in volume of sphere one (ΔV1) can be determined using the equation for bulk modulus:
Bulk modulus (B) = -(Δp/ΔV)/V0
Where Δp is the change in pressure, ΔV is the change in volume, and V0 is the original volume.
Since we are increasing the pressure on both spheres by the same amount, the change in pressure (Δp) will be the same for both spheres. Therefore, we can ignore the term Δp in the equation.
For sphere two, the bulk modulus (B2) is twice as large as the bulk modulus of sphere one (B1). Therefore, we can write the relationship between the bulk moduli as:
B2 = 2*B1
Now let's consider the relationship between the change in volume and the bulk modulus. We can rearrange the bulk modulus equation to solve for ΔV:
ΔV = (-Δp/B)*V0
Substituting the relationship between the bulk moduli, we have:
ΔV1 = (-Δp/B1)*V0
ΔV2 = (-Δp/B2)*V0
Now we can compare the two equations:
ΔV2 = (-Δp/B2)*V0
= (-Δp/(2*B1))*V0
= (1/2)*((-Δp/B1)*V0)
= (1/2)*ΔV1
Therefore, the change in volume of sphere two (ΔV2) is equal to half the change in volume of sphere one (ΔV1).
So the correct answer is:
D) ΔV2 = 1/2ΔV1