I'll solve this problem about the bulk modulus of ice.
The question provides the bulk modulus of ice as $2.20 \times 10^{9} \mathrm{N} / \mathrm{m}^{2}$. Let's analyze what this means.
Understanding bulk modulus
The bulk modulus (B) is a measure of a substance's resistance to uniform compression. It is defined as the ratio of the pressure change (ΔP) to the resulting relative change in volume (ΔV/V):
\( B = -V \frac{\Delta P}{\Delta V} \)
The negative sign indicates that an increase in pressure causes a decrease in volume.
For ice, \( B = 2.20 \times 10^{9} \mathrm{N/m^2} = 2.20 \mathrm{~GPa} \)
Interpreting the physical meaning
This value tells us how much pressure is needed to cause a relative volume change in ice. A higher bulk modulus indicates a more rigid material that resists compression.
For comparison:
- Water at room temperature has a bulk modulus of about 2.2 GPa
- Steel has a bulk modulus of about 160 GPa
- Rubber has a bulk modulus of about 1.5 GPa
So ice is relatively incompressible (similar to liquid water) but much less rigid than metals like steel.
Calculating volume change under pressure
If we apply a pressure of 1 MPa (10^6 N/m²) to ice, we can calculate the fractional volume change:
\( \frac{\Delta V}{V} = -\frac{\Delta P}{B} = -\frac{10^6 \mathrm{~N/m^2}}{2.20 \times 10^9 \mathrm{~N/m^2}} = -4.55 \times 10^{-4} \)
This means the volume would decrease by about 0.0455% under this pressure.
\(\boxed{\text{The bulk modulus of ice is } 2.20 \times 10^9 \mathrm{~N/m^2}, \text{ indicating it is relatively incompressible.}}\)
\(\boxed{\text{The bulk modulus of ice is } 2.20 \times 10^9 \mathrm{~N/m^2}, \text{ indicating it is relatively incompressible.}}\)
Answered
