Silicon for computer chips is grown in large cylinders called "boules" that are \(300 \mathrm{~mm}\) in diameter and \(2 \mathrm{~m}\) in length, as shown. The density of silicon is \(2.33 \mathrm{~g} / \mathrm{cm}^{3}\). Silicon wafers for making integrated circuits are sliced from a \(2.0-\mathrm{m}\) boule and are typically \(0.75 \mathrm{~mm}\) thick and \(300 \mathrm{~mm}\) in diameter. (a) How many wafers can be cut from a single boule? (b) What is the mass of a silicon wafer? (The volume of a cylinder is given by \(\pi r^{2} h,\) where \(r\) is the radius and \(h\) is its height.)

We have a silicon boule with a length of 2m, and we want to slice wafers with a thickness of 0.75mm. To find out how many wafers can be obtained, we need to divide the length of the boule by the thickness of a wafer. First, convert the length of the boule and the thickness of the wafer to the same unit, centimeters: \[length_{boule} = 2m \times 100cm/m = 200cm\] \[thickness_{wafer} = 0.75mm \times 0.1cm/mm = 0.075cm\] Now, we can calculate the number of wafers: \[number\_of\_wafers = \frac{length_{boule}}{thickness_{wafer}}\]

We have the diameter and the thickness of the wafer, so we can calculate its volume. We know that the volume of a cylinder is given by the formula \(\pi r^{2}h\), where \(r\) is the radius, and \(h\) is the height (thickness). First, calculate the radius of the wafer: \[radius_{wafer} = \frac{diameter_{wafer}}{2} = \frac{300mm}{2} \times 0.1cm/mm = 15cm\] Now, we can calculate the volume: \[volume_{wafer} = \pi \times (radius_{wafer})^{2} \times thickness_{wafer}\]

The mass of a silicon wafer can be calculated by multiplying the volume of the wafer by the density of silicon. We have the volume of the wafer from the previous step, and we know that the density of silicon is 2.33 g/cm³: \[mass_{wafer} = density \times volume_{wafer}\] Now, put everything together and make the calculations: (a) Number of wafers \[number\_of\_wafers = \frac{200cm}{0.075cm} \approx 2667\] (b) Mass of a silicon wafer \[volume_{wafer} = \pi \times (15cm)^{2} \times 0.075cm\] \[volume_{wafer} \approx 53.03 cm^3\] \[mass_{wafer} = 2.33 g/cm³ \times 53.03 cm^3\] \[mass_{wafer} \approx 123.55g\] So, (a) we can cut approximately 2667 wafers from a single boule, and (b) the mass of a silicon wafer is approximately 123.55g.