Problem 15 - College Physics

Question

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

Pure water: 1000 g, Whole blood: 1050 g, Seawater: 1030 g.

Step 1: Understand the Relationship between Density, Mass, and Volume

Density is defined as mass per unit volume. The formula is given by \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This means that mass can be calculated if the density and volume are known using \( \text{Mass} = \text{Density} \times \text{Volume} \).

Step 2: Convert Volume to Appropriate Units

The volume given is 1.00 liters (L). Since density is given in \( \mathrm{g/cm}^3 \), it’s helpful to express volume in \( \mathrm{cm}^3 \). We know that \( 1.00 \mathrm{~L} = 1000 \mathrm{~cm}^3 \).

Step 3: Calculate Mass of Pure Water

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

Step 4: Calculate Mass of Whole Blood

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

Step 5: Calculate Mass of Seawater

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

Key Concepts

Mass Calculation

To determine the mass of a substance when you know its density and volume, you use the formula \( \text{Mass} = \text{Density} \times \text{Volume} \). This simple equation is extremely helpful in many scientific fields. If you have a liquid with a known density and you measure how much space it takes up (volume), you can calculate its mass:
\[ \text{Mass} = \text{Density} \times \text{Volume} \]
For example:

• By multiplying the density of pure water \(1.00 \, \text{g/cm}^3\) with its volume \(1000 \, \text{cm}^3\), we find the mass of pure water to be \(1000 \, \text{g}\).
• For a denser substance like blood, with a density of \(1.05 \, \text{g/cm}^3\), its mass calculates to \(1050 \, \text{g}\).
• Seawater, at \(1.03 \, \text{g/cm}^3\), results in a mass of \(1030 \, \text{g}\) using the same volume.
  

This process ensures that we accurately calculate mass regardless of the substance's density.

Volume Conversion

Converting units of volume is essential, especially when different units are used in the given data. Volume conversion is about changing a liquid's measured space from one unit to another. In this problem, we convert liters to cubic centimeters to match the density units of \( \text{g/cm}^3\).

When you have a volume in liters, remember:

• \(1.00 \, \text{L} = 1000 \, \text{cm}^3\).
  

This means whenever you have a volume expressed in liters and need to convert it to cubic centimeters, simply multiply by 1000. Understanding how to switch between these two units allows you to use density measurements that often rely on \( \text{cm}^3 \).

Unit Conversion

Unit conversion is a key skill in scientific problem solving. Different systems of units are often used, such as the metric system for scientific measurements. In this specific exercise, we need to convert units to apply the correct formula.

• The volume was initially given in liters, a common unit for larger quantities of liquids.
• For calculation purposes, it was converted to cubic centimeters \( \text{cm}^3 \) to align with the density unit \( \text{g/cm}^3 \).
  

Correct unit conversion involves carefully keeping track of these differences. It ensures that the math accurately reflects the situation you're analyzing, preventing errors that come from mismatched units.

Fluid Densities

Understanding fluid densities is crucial in fields like chemistry, physics, and engineering. Density indicates how much mass is contained in a specific volume and varies between different substances.

• For pure water, the density is \(1.00 \, \text{g/cm}^3\), which is a standard reference point.
• Whole blood is slightly more dense at \(1.05 \, \text{g/cm}^3\), indicating it has more mass per unit volume than water.
• Seawater, with a density of \(1.03 \, \text{g/cm}^3\), is denser than pure water due to dissolved salts.
  

These differences in density affect how substances behave and interact in different contexts, such as floating or mixing.

Step-by-Step Solution

Breaking down a problem into a step-by-step solution can clarify complex concepts. In this exercise, steps guide the process from understanding the relationship between density, mass, and volume, to converting units, and finally calculating the mass for each fluid:

• Step 1: Recognize the formula \( \text{Mass} = \text{Density} \times \text{Volume} \).
• Step 2: Convert 1 L to \(1000 \, \text{cm}^3\).
• Step 3-5: Calculate mass for water, blood, and seawater.
  

By following these structured steps, students can systematically tackle similar problems, simplifying the calculation processes and enhancing comprehension.