Artificial diamonds can be made using high-pressure, high-temperature presses. Suppose an artificial diamond of volume 1.30 x 10-6 m3 is formed under a pressure of 4.70 GPa. Find the change in its volume (in m³) when it is released from the press and brought to atmospheric pressure. Take the diamond's bulk modulus to be B = 194 GPa. (Assume the system is at sea level.)
Artificial diamonds can be made using high-pressure, high-temperature presses. Suppose an artificial diamond of volume 1.30 x 10-6 m3 is formed under a pressure of 4.70 GPa. Find the change in its volume (in m³) when it is released from the press and brought to atmospheric pressure. Take the diamond's bulk modulus to be B = 194 GPa. (Assume the system is at sea level.)
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![### Problem Explanation: Volume Change in Artificial Diamond Due to Pressure Difference
Artificial diamonds can be created by using high-pressure, high-temperature presses. Suppose an artificial diamond with a volume of \(1.30 \times 10^{-6} \, \text{m}^3\) is formed under a pressure of \(4.70 \, \text{GPa}\).
#### Objective:
Find the change in its volume (\( \Delta V \) in \( \text{m}^3 \)) when it is released from the press and brought to atmospheric pressure.
#### Given Data:
- Initial volume, \( V = 1.30 \times 10^{-6} \, \text{m}^3 \)
- Pressure under which it is formed, \( P = 4.70 \, \text{GPa} \)
- Bulk modulus of diamond, \( B = 194 \, \text{GPa} \)
- Atmospheric pressure is assumed to be negligible compared to the pressure under which the diamond was formed.
#### Approach to Solution:
1. **Understand Bulk Modulus**:
- The bulk modulus \( B \) is a measure of a substance's resistance to uniform compression.
- It is given by the formula:
\[
B = -\frac{\Delta P}{\frac{\Delta V}{V}}
\]
Where \( \Delta P \) is the change in pressure and \( \frac{\Delta V}{V} \) is the relative change in volume.
2. **Calculate the Change in Volume**:
- Rearrange the bulk modulus formula to solve for \( \Delta V \):
\[
\frac{\Delta V}{V} = -\frac{\Delta P}{B}
\]
3. **Substitute Given Values**:
- \( \Delta P \approx 4.70 \, \text{GPa} \) (since atmospheric pressure is negligible)
- \( B = 194 \, \text{GPa} \)
- Calculate \( \frac{\Delta V}{V} \):
\[
\frac{\Delta V}{V} = -\frac{4.70}{194}
\]
\[
\frac{\Delta V}{V} = -0.0242
\]
- Thus,
\[
\Delta V = V](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F32cd56cf-4bd6-4dcd-a322-7126cf104832%2F9cb0bb11-2948-4150-99d3-a499ad1ace77%2Fmu4c4yb.jpeg&w=3840&q=75)
Transcribed Image Text:### Problem Explanation: Volume Change in Artificial Diamond Due to Pressure Difference
Artificial diamonds can be created by using high-pressure, high-temperature presses. Suppose an artificial diamond with a volume of \(1.30 \times 10^{-6} \, \text{m}^3\) is formed under a pressure of \(4.70 \, \text{GPa}\).
#### Objective:
Find the change in its volume (\( \Delta V \) in \( \text{m}^3 \)) when it is released from the press and brought to atmospheric pressure.
#### Given Data:
- Initial volume, \( V = 1.30 \times 10^{-6} \, \text{m}^3 \)
- Pressure under which it is formed, \( P = 4.70 \, \text{GPa} \)
- Bulk modulus of diamond, \( B = 194 \, \text{GPa} \)
- Atmospheric pressure is assumed to be negligible compared to the pressure under which the diamond was formed.
#### Approach to Solution:
1. **Understand Bulk Modulus**:
- The bulk modulus \( B \) is a measure of a substance's resistance to uniform compression.
- It is given by the formula:
\[
B = -\frac{\Delta P}{\frac{\Delta V}{V}}
\]
Where \( \Delta P \) is the change in pressure and \( \frac{\Delta V}{V} \) is the relative change in volume.
2. **Calculate the Change in Volume**:
- Rearrange the bulk modulus formula to solve for \( \Delta V \):
\[
\frac{\Delta V}{V} = -\frac{\Delta P}{B}
\]
3. **Substitute Given Values**:
- \( \Delta P \approx 4.70 \, \text{GPa} \) (since atmospheric pressure is negligible)
- \( B = 194 \, \text{GPa} \)
- Calculate \( \frac{\Delta V}{V} \):
\[
\frac{\Delta V}{V} = -\frac{4.70}{194}
\]
\[
\frac{\Delta V}{V} = -0.0242
\]
- Thus,
\[
\Delta V = V
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