Assuming that the volume stays constant, how much an increase in surface area will result from a spherical gold particle of 5 cm in diameter if it is reduced to a group of gold nanoparticles with 5 nm diameter?
Calculation of Increase in Surface Area
Data:
- Diameter of gold particle = 5 cm, so radius = 2.5 cm
- Diameter of gold nanoparticle = 5 nm, so radius = 2.5 nm
Solution:
To calculate the increase in surface area, we use the formula A = 4πr^2 for the gold particle and the nanoparticles. The volume conservation determines the number of nanoparticles obtained. Finally, we multiply the surface area of one nanoparticle by the total number to find the total surface area of the nanoparticles.
Explanation:
To calculate the increase in surface area, we need to find the surface area of the spherical gold particle and compare it to the total surface area of the gold nanoparticles. The surface area of a sphere is given by the formula A = 4πr^2, where r is the radius of the sphere. For the gold particle with a diameter of 5 cm (radius of 2.5 cm), the surface area is A = 4π(2.5)^2 = 4π(6.25) = 25π cm^2.
When the gold particle is reduced to gold nanoparticles with a diameter of 5 nm (radius of 2.5 nm), the total surface area of the nanoparticles is the sum of the surface areas of each nanoparticle. Assuming the nanoparticles are evenly distributed, we can use the formula A = 4πr^2 to calculate the surface area of one nanoparticle, and then multiply it by the total number of nanoparticles.
However, it is important to note that the number of nanoparticles obtained from the original gold particle depends on the volume conservation. Since the volume is constant, we have (4/3)πr^3 = (4/3)π(2.5)^3 = (4/3)π(15.625) cm^3. Dividing this volume by the volume of one nanoparticle (4/3)π(0.0125)^3 = (4/3)π(1.56 x 10^-6) cm^3, we can find the total number of nanoparticles.
Once we have the total number of nanoparticles, we can calculate the surface area of one nanoparticle and multiply it by the total number to find the total surface area of the nanoparticles.