The density of titanium metal is 4.506 g/cm³ at 20 °C. What is the mass of a spherical ball of titanium with a diameter of 2.00 cm. [The volume of a sphere is (4/3) πr³, where r is the radius.]

Verified step by step guidance

1. <Calculate the radius of the sphere by dividing the diameter by 2: r = \frac{2.00 \text{ cm}}{2}.>
2. <Use the formula for the volume of a sphere: V = \frac{4}{3} \pi r^3, substituting the radius you calculated.>
3. <Calculate the volume of the sphere using the value of \pi \approx 3.14159.>
4. <Use the density formula, \text{Density} = \frac{\text{Mass}}{\text{Volume}}, to solve for mass: \text{Mass} = \text{Density} \times \text{Volume}.>
5. <Substitute the given density of titanium and the calculated volume into the formula to find the mass of the titanium sphere.>

Key Concepts

Density

Density is defined as mass per unit volume, typically expressed in grams per cubic centimeter (g/cm³) for solids. It is a crucial property that helps determine how much mass is contained in a given volume of a substance. In this problem, the density of titanium is provided, which will be used to calculate the mass of the spherical ball once its volume is determined.

Volume of a Sphere

The volume of a sphere can be calculated using the formula V = (4/3)πr³, where r is the radius of the sphere. This formula is essential for determining how much space the spherical ball occupies. In this case, the diameter is given, so the radius must be calculated by dividing the diameter by two before applying the formula.

Mass Calculation

Mass can be calculated using the formula mass = density × volume. Once the volume of the titanium sphere is determined using the volume formula, this relationship allows us to find the mass by multiplying the volume by the density of titanium. This step is critical for solving the problem and finding the mass of the spherical ball.