Problem 7 - Physics

Question

A cylindrical glass of water \(\left(\mathrm{H}_{2} \mathrm{O}\right)\) has a radius of \(4.50 \mathrm{~cm}\) and a height of \(12.0 \mathrm{~cm} .\) The density of water is \(1.00 \mathrm{~g} / \mathrm{cm}^{3}\). How many moles of water molecules are contained in the glass?

Step-by-Step Solution

Answer

The glass contains approximately 42.38 moles of water molecules.

Step 1: Calculate the Volume of Water

To find the volume of water in the cylindrical glass, use the formula for the volume of a cylinder: \[ V = \pi r^2 h \]Substitute the given values of the radius \( r = 4.50 \text{ cm} \) and height \( h = 12.0 \text{ cm} \):\[ V = \pi \times (4.50)^2 \times 12.0 \approx 763.41 \text{ cm}^3 \]

Step 2: Calculate the Mass of Water

With the volume known, use the density of water to find its mass:\[ \text{Mass} = \text{Density} \times \text{Volume} \]Given \( \text{Density} = 1.00 \text{ g/cm}^3 \), substitute the volume from step 1:\[ \text{Mass} = 1.00 \times 763.41 \approx 763.41 \text{ g} \]

Step 3: Calculate the Number of Moles of Water

Finally, to find the number of moles, use the formula:\[ \text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}} \]The molar mass of water \( \mathrm{H}_2 \mathrm{O} \) is \( 18.015 \text{ g/mol} \). Substitute the mass from step 2:\[ \text{Moles} = \frac{763.41}{18.015} \approx 42.38 \text{ moles} \]

Key Concepts

Cylindrical Volume Calculation

To understand how to find the amount of water in a cylindrical glass, it's essential to first calculate the volume of the cylinder. A cylinder is a 3D shape with two parallel circles at the ends. The volume measures how much space the substance like water inside the glass occupies.
The formula for the volume of a cylinder is \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height. For our glass, the radius is given as \(4.50 \text{ cm}\) and the height is \(12.0 \text{ cm}\). Substitute these values into the formula:

• Calculate \(r^2 = (4.50 \text{ cm})^2 = 20.25 \text{ cm}^2\).
• Multiply by \(\pi\): \(\pi \times 20.25 \approx 63.62 \text{ cm}^2\).
• Finally, multiply by the height \(h\), so \(V = 63.62 \times 12.0 \approx 763.41 \text{ cm}^3\).

This calculation tells us that the volume of water that fits into this cylindrical glass is approximately \(763.41 \text{ cm}^3\).

Density and Mass Relationship

Once you have the volume, the next step is finding the mass of the water. Mass is how much matter there is in the water, and it depends on both the volume and the density of the water. The density is a measure of mass per unit of volume and tells us how tightly packed the water molecules are.
To find the mass, you use the formula \(\text{Mass} = \text{Density} \times \text{Volume}\). For water, the density is typically \(1.00 \text{ g/cm}^3\). This means every cubic centimeter of water has a mass of 1 gram.
Using the volume from the previous section:

• The calculation becomes \(\text{Mass} = 1.00 \times 763.41 \text{ cm}^3 = 763.41 \text{ g}\).

This result indicates the water in the glass weighs approximately \(763.41 \text{ g}\).

Molar Mass of Water

To convert the mass of the water into moles, it's crucial to understand the concept of molar mass. Molar mass is the mass of a substance divided by the amount of substance, in units of grams per mole. It represents the mass of one mole (6.022 \(\times\) 10\(^ {23}\) molecules) of that substance. For water, the molar mass is \(18.015 \text{ g/mol}\).
Using the formula \(\text{Moles} = \frac{\text{Mass}}{\text{Molar Mass}}\), you can find the moles of water:

• Plug in the mass from the previous section and the molar mass of water: \(\text{Moles} = \frac{763.41}{18.015}\).
• This results in approximately \(42.38\) moles of water.

Understanding moles helps us quantify the number of molecules in the water, connecting mass to the actual number of molecules in the sample.