Problem 105 - General Chemistry: Principles and Modern Applications

Question

The density of water is \(0.9982 \mathrm{g} / \mathrm{cm}^{3}\) at \(20^{\circ} \mathrm{C}\). Express the density of water at \(20^{\circ} \mathrm{C}\) in the following units: (a) \(\mathrm{g} / \mathrm{L} ;\) (b) \(\mathrm{kg} / \mathrm{m}^{3} ;\) (c) \(\mathrm{kg} / \mathrm{km}^{3}\).

Step-by-Step Solution

Answer

The density of water is: (a) 998.2 g/L, (b) 998.2 kg/m³, and (c) \(9.982 * 10^{11}\) kg/km³

Step 1: Convert to Grams per Litre (g/L)

First, remember that 1 litre(I), is equal to \(1000 cm^{3}\). So, to convert the density to grams per litre, multiply the original density by the number of cubic centimetres in a litre: \(0.9982 g/cm^{3} * 1000 cm^{3}/L = 998.2 g/L\)

Step 2: Convert to Kilograms per Cubic Meter (kg/m³)

Next, recall that 1 kilogram (kg) is equal to 1000 grams (g) and 1 meter (m) is equivalent to 100 centimeters (cm). Therefore, \(1 g = 0.001 kg\) and \(1 cm = 0.01 m\), so a cubic meter is equivalent to \(100^{3} cm^{3} = 1 000 000 cm^{3}\). The conversion therefore becomes: \(0.9982 g/cm^{3} * 0.001 kg/g * 1 000 000 cm^{3}/m^{3} = 998.2 kg/m^{3}\)

Step 3: Convert to Kilograms per Cubic Kilometer (kg/km³)

Lastly, consider that 1 kilometer (km) is equivalent to 1000 meters (m). Therefore a cubic kilometer is equivalent to \(1000^{3} m^{3} = 1 000 000 000 m^{3}\). The conversion therefore becomes: \(998.2 kg/m^{3} * 1 000 000 000 m^{3}/km^{3} = 9.982 * 10^{11} kg/km^{3}\)

Key Concepts

Grams per litre

The concept of density expressed in grams per litre (g/L) is often used when dealing with liquids, such as water. It's quite straightforward to understand and visualize. When you measure density in grams per litre, you're essentially asking how many grams of a substance are contained in every litre of volume. For water, the conversion from grams per cubic centimetre (g/cm³) uses the fact that 1 litre equals 1000 cubic centimetres. This means you simply multiply the density in g/cm³ by 1000 to get the density in g/L.

In our example, water has a density of 0.9982 g/cm³, which after conversion equals 998.2 g/L. This indicates that about 998.2 grams of water fill one litre at 20°C.

• 1 g/cm³ = 1000 g/L
• Density of water at 20°C = 998.2 g/L
• Uses: Common in Chemistry and Biology

Kilograms per cubic meter

Density indicated in kilograms per cubic meter (kg/m³) is typically used in physics and engineering. This unit measures how much mass exists in a contained area of one cubic meter. Converting from grams per cubic centimetre to kilograms per cubic meter involves two straightforward steps:

• First, convert grams to kilograms by knowing that there are 1000 grams in a kilogram.
• Then, scale up from cubic centimetre to cubic meter, keeping in mind that 1 cubic meter equals 1,000,000 cubic centimetres.

In the given example, the water's density translates to 998.2 kg/m³. This conversion accurately shows the mass of the water if you have a one-meter cube filled with it.

• 1 g/cm³ = 1000 kg/m³
• Density of water at 20°C = 998.2 kg/m³
• Uses: Engineering, Physics

Kilograms per cubic kilometer

When density is expressed in kilograms per cubic kilometer (kg/km³), it represents an incredibly large volume. It's a less common unit for density but can be used in scientific fields dealing with large-scale geophysical phenomena. Understanding this conversion involves expanding from cubic meters to cubic kilometers.

With 1 kilometer equaling 1000 meters, a cubic kilometer is equivalent to 1,000,000,000 cubic meters. Therefore, to find the density in kg/km³, you multiply the density in kg/m³ by 1,000,000,000.

In our scenario, water's density translates to approximately 9.982 * 10^{11} kg/km³. This massive number reflects the vast volume involved when considering an entire cubic kilometer of water.

• 1 kg/m³ = 1,000,000,000 kg/km³
• Density of water at 20°C = 9.982 * 10^{11} kg/km³
• Uses: Geophysics, Environmental Science