Answer:
The volume percent of graphite is 91.906 per cent.
Explanation:
The volume percent of graphite (
) is determined by the following expression:
![\%V_{Gr} = \frac{V_{Gr}}{V_{Gr}+V_{Fe}} \times 100\,\%](https://tex.z-dn.net/?f=%5C%25V_%7BGr%7D%20%3D%20%5Cfrac%7BV_%7BGr%7D%7D%7BV_%7BGr%7D%2BV_%7BFe%7D%7D%20%5Ctimes%20100%5C%2C%5C%25)
![\%V_{Gr} = \frac{1}{1+\frac{V_{Gr}}{V_{Fe}} }\times 100\,\%](https://tex.z-dn.net/?f=%5C%25V_%7BGr%7D%20%3D%20%5Cfrac%7B1%7D%7B1%2B%5Cfrac%7BV_%7BGr%7D%7D%7BV_%7BFe%7D%7D%20%7D%5Ctimes%20100%5C%2C%5C%25)
Where:
- Volume occupied by the graphite phase, measured in cubic centimeters.
- Volume occupied by the ferrite phase, measured in cubic centimeters.
The volume of each phase can be calculated in terms of its density and mass. That is:
![V_{Gr} = \frac{m_{Gr}}{\rho_{Gr}}](https://tex.z-dn.net/?f=V_%7BGr%7D%20%3D%20%5Cfrac%7Bm_%7BGr%7D%7D%7B%5Crho_%7BGr%7D%7D)
![V_{Fe} = \frac{m_{Fe}}{\rho_{Fe}}](https://tex.z-dn.net/?f=V_%7BFe%7D%20%3D%20%5Cfrac%7Bm_%7BFe%7D%7D%7B%5Crho_%7BFe%7D%7D)
Where:
,
- Masses of the graphite and ferrite phases, measured in grams.
,
- Densities of the graphite and ferrite phases, measured in grams per cubic centimeter.
Let substitute each volume in the definition of the volume percent of graphite:
![\%V_{Gr} = \frac{1}{1 +\frac{\frac{m_{Gr}}{\rho_{Gr}} }{\frac{m_{Fe}}{\rho_{Fe}} } } \times 100\,\%](https://tex.z-dn.net/?f=%5C%25V_%7BGr%7D%20%3D%20%5Cfrac%7B1%7D%7B1%20%2B%5Cfrac%7B%5Cfrac%7Bm_%7BGr%7D%7D%7B%5Crho_%7BGr%7D%7D%20%7D%7B%5Cfrac%7Bm_%7BFe%7D%7D%7B%5Crho_%7BFe%7D%7D%20%7D%20%7D%20%5Ctimes%20100%5C%2C%5C%25)
![\%V_{Gr} = \frac{1}{1+\left(\frac{m_{Gr}}{m_{Fe}} \right)\cdot \left(\frac{\rho_{Fe}}{\rho_{Gr}} \right)}\times 100\,\%](https://tex.z-dn.net/?f=%5C%25V_%7BGr%7D%20%3D%20%5Cfrac%7B1%7D%7B1%2B%5Cleft%28%5Cfrac%7Bm_%7BGr%7D%7D%7Bm_%7BFe%7D%7D%20%5Cright%29%5Ccdot%20%5Cleft%28%5Cfrac%7B%5Crho_%7BFe%7D%7D%7B%5Crho_%7BGr%7D%7D%20%5Cright%29%7D%5Ctimes%20100%5C%2C%5C%25)
Let suppose that 100 grams of cast iron are available, masses of each phase are now determined:
![m_{Gr} = \frac{2.5}{100}\times (100\,g)](https://tex.z-dn.net/?f=m_%7BGr%7D%20%3D%20%5Cfrac%7B2.5%7D%7B100%7D%5Ctimes%20%28100%5C%2Cg%29)
![m_{Gr} = 2.5\,g](https://tex.z-dn.net/?f=m_%7BGr%7D%20%3D%202.5%5C%2Cg)
![m_{Fe} = 100\,g - 2.5\,g](https://tex.z-dn.net/?f=m_%7BFe%7D%20%3D%20100%5C%2Cg%20-%202.5%5C%2Cg)
![m_{Fe} = 97.5\,g](https://tex.z-dn.net/?f=m_%7BFe%7D%20%3D%2097.5%5C%2Cg)
If
,
,
and
, the volume percent of graphite is:
![\%V_{Gr} = \frac{1}{1+\left(\frac{2.5\,gr}{97.5\,gr} \right)\cdot \left(\frac{7.9\,\frac{g}{cm^{3}} }{2.3\,\frac{g}{cm^{3}} } \right)} \times 100\,\%](https://tex.z-dn.net/?f=%5C%25V_%7BGr%7D%20%3D%20%5Cfrac%7B1%7D%7B1%2B%5Cleft%28%5Cfrac%7B2.5%5C%2Cgr%7D%7B97.5%5C%2Cgr%7D%20%5Cright%29%5Ccdot%20%5Cleft%28%5Cfrac%7B7.9%5C%2C%5Cfrac%7Bg%7D%7Bcm%5E%7B3%7D%7D%20%7D%7B2.3%5C%2C%5Cfrac%7Bg%7D%7Bcm%5E%7B3%7D%7D%20%7D%20%5Cright%29%7D%20%5Ctimes%20100%5C%2C%5C%25)
![\% V_{Gr} = 91.906\,\%](https://tex.z-dn.net/?f=%5C%25%20V_%7BGr%7D%20%3D%2091.906%5C%2C%5C%25)
The volume percent of graphite is 91.906 per cent.