Problem 18 - College Physics

Question

\(\cdot\) Blood is thicker than water. The density (mass divided by volume) of pure water is \(1.00 \mathrm{g} / \mathrm{cm}^{3},\) that of whole blood is \(1.05 \mathrm{g} / \mathrm{cm}^{3},\) and the density of seawater is 1.03 \(\mathrm{g} / \mathrm{cm}^{3} .\) What is the mass (in grams) of 1.00 \(\mathrm{L}\) of each of these substances?

Step-by-Step Solution

Answer

Mass of 1 L of pure water is 1000 g, whole blood is 1050 g, and seawater is 1030 g.

Step 1: Understand the Equation for Mass Calculation

The formula to calculate the mass of a substance is given by the product of its volume and density. This can be expressed as \( \text{mass} = \text{density} \times \text{volume} \). This formula will be used to calculate the mass for each of the substances: pure water, whole blood, and seawater.

Step 2: Convert Volume from Liters to Cubic Centimeters

Since the densities of the substances are given in grams per cubic centimeter (\( \mathrm{g/cm}^3 \)), we first need to convert the volume from liters to cubic centimeters. Given that \(1\, \text{L} = 1000\, \text{cm}^3\), the volume for each substance is \(1000\, \text{cm}^3\).

Step 3: Calculate the Mass of Pure Water

Using the density of pure water (\(1.00\, \mathrm{g/cm}^3\)) and the volume (\(1000\, \text{cm}^3\)), the mass of pure water is calculated as follows:\[\text{mass} = 1.00\, \mathrm{g/cm}^3 \times 1000\, \text{cm}^3 = 1000\, \text{g}\]

Step 4: Calculate the Mass of Whole Blood

Using the density of whole blood (\(1.05\, \mathrm{g/cm}^3\)) and the volume (\(1000\, \text{cm}^3\)), the mass of whole blood is calculated as follows:\[\text{mass} = 1.05\, \mathrm{g/cm}^3 \times 1000\, \text{cm}^3 = 1050\, \text{g}\]

Step 5: Calculate the Mass of Seawater

Using the density of seawater (\(1.03\, \mathrm{g/cm}^3\)) and the volume (\(1000\, \text{cm}^3\)), the mass of seawater is calculated as follows:\[\text{mass} = 1.03\, \mathrm{g/cm}^3 \times 1000\, \text{cm}^3 = 1030\, \text{g}\]

Key Concepts

Mass Calculation

To determine the mass of various substances, such as water, blood, and seawater, we use the equation: \[\text{Mass} = \text{Density} \times \text{Volume}\] This means that the mass of a substance can be calculated by multiplying its density by its volume. It's a simple yet powerful tool in physics and chemistry. Mass calculation helps us understand how much of a substance is contained within a certain volume. - **Density** is the measure of how much mass a substance contains in a given volume, typically expressed in grams per cubic centimeter (g/cm³).- **Volume** is simply the amount of space that a substance occupies, typically measured in liters (L) or cubic centimeters (cm³).Whenever you need to find the mass of a substance, ensure you have both its density and volume ready for this straightforward calculation.

Volume Conversion

Before performing any mass calculation with densities given in grams per cubic centimeter (g/cm³), it is crucial to convert the volume from liters to cubic centimeters. This conversion is necessary because we want to match the units for density and volume. Since 1 liter (L) is equivalent to 1000 cubic centimeters (cm³), this conversion is simple: - If your substance is measured in liters, multiply by 1000 to convert to cubic centimeters. This step ensures consistency and accuracy in calculations, preventing errors caused by mismatched units.

Density of Water

The density of pure water is precisely 1.00 g/cm³ at standard temperature and pressure (STP). This consistency makes it a standard in scientific measurements. Calculating the mass of water using its density is straightforward. For example, if you have 1 liter of water:- Convert 1 L to 1000 cm³.- Multiply the volume by the density to find the mass: \[\text{Mass} = 1.00\, \mathrm{g/cm}^3 \times 1000\, \mathrm{cm}^3 = 1000\, \mathrm{g}\]It confirms that 1 liter of water corresponds to 1000 grams, acting as a convenient reference point for other fluids.

Density of Blood

Blood is slightly denser than water, which is why its density is 1.05 g/cm³. Understanding the density of blood is crucial in the medical field for various diagnostics and treatment plans. Here's how you calculate the mass for 1 liter of blood:- Convert 1 L to 1000 cm³.- Multiply the volume by the density:\[\text{Mass} = 1.05\, \mathrm{g/cm}^3 \times 1000\, \mathrm{cm}^3 = 1050\, \mathrm{g}\]The fact that 1 liter of blood weighs more than 1 liter of water is essential when considering blood volume in clinical settings.

Density of Seawater

Seawater is slightly denser than pure water due to the salts dissolved in it, giving it a density of 1.03 g/cm³. This difference plays a significant role in oceanography, maritime navigation, and environmental science.To calculate the mass of 1 liter of seawater:- Convert the volume from liters to cubic centimeters: 1 L equals 1000 cm³.- Use the density to find the mass:\[\text{Mass} = 1.03\, \mathrm{g/cm}^3 \times 1000\, \mathrm{cm}^3 = 1030\, \mathrm{g}\]The increased mass of seawater compared to fresh water is essential for understanding buoyancy and the behavior of sea life.