91Ó°ÊÓ

To determine the volume of a cylindrical glass of water, we need to use the formula for the volume of a cylinder. This formula is:\[ V = \pi r^2 h \]where:

• \( V \) is the volume.
• \( r \) is the radius of the base.
• \( h \) is the height of the cylinder.

In this exercise, the glass of water has a radius \( r = 4.50 \text{ cm} \) and a height \( h = 12.0 \text{ cm} \). Substituting these values into the formula gives:\[ V = \pi (4.5)^2 (12.0) \]First, square the radius: \( (4.5)^2 = 20.25 \). Then multiply by the height \( 12.0 \):\[ 20.25 \times 12.0 = 243.0 \]Finally, multiply this by \( \pi \):\[ V = 243.0 \times \pi \approx 763.407 \text{ cm}^3 \]Thus, the volume of the cylinder is approximately \( 763.407 \text{ cm}^3 \).

Density is an important concept in physics and chemistry because it relates the mass of a substance to its volume. For water, the density is typically given as \( 1.00 \text{ g/cm}^3 \). This means each cubic centimeter of water has a mass of 1 gram.To find the mass of water in the glass, we use the volume calculated earlier. The formula to find mass from volume and density is:\[ m = V \times \text{density} \]Substitute the previously calculated volume \( V \approx 763.407 \text{ cm}^3 \) into the formula:\[ m = 763.407 \times 1.00 = 763.407 \text{ g} \]Therefore, the mass of water in the glass is approximately \( 763.407 \text{ g} \). This step is crucial because it provides the mass needed to convert to moles.

Converting mass to moles is a common calculation in chemistry that allows us to understand how many particles, such as molecules, are present in a given mass. With water, this involves using the molar mass. The number of moles \( n \) can be found by dividing the mass \( m \) by the molar mass \( M \):\[ n = \frac{m}{M} \]In this case, the mass \( m \) of the water is approximately \( 763.407 \text{ g} \), as calculated previously. The molar mass of water is \( 18.02 \text{ g/mol} \).Substitute the values into the equation:\[ n = \frac{763.407}{18.02} \approx 42.38 \text{ moles} \]Hence, the glass contains approximately \( 42.38 \text{ moles} \) of water. This tells us the amount of water molecules in the glass, using Avogadro’s number for conversions to the actual number of molecules later.

Understanding molar mass is crucial in chemistry, as it enables the transition from a macroscopic scale to a microscopic scale by linking grams to moles. For water (\( \text{H}_2\text{O} \)), the molar mass is derived from the atomic masses of hydrogen and oxygen.Water consists of two hydrogen atoms and one oxygen atom. The atomic mass of hydrogen is approximately \( 1.01 \text{ g/mol} \), and for oxygen, it's about \( 16.00 \text{ g/mol} \). Therefore, the molar mass \( M \) of water is calculated as follows:\[ M = (2 \times 1.01) + 16.00 = 18.02 \text{ g/mol} \]This means that one mole of water weighs \( 18.02 \text{ g} \), an essential piece of information when calculating moles from given mass. This molar mass allows us to relate the mass of a sample to the number of molecules it contains through conversions with Avogadro's number.