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Find the density and specific gravity of gasoline if 51 g occupies 75 \(\mathrm{cm}^{3}\). Make sure you know how to convert cubic centimeters to cubic meters: \(1.0 \mathrm{~m}^{3}=1.0 \times 10^{6} \mathrm{~cm}^{3}\). 1 $$ \begin{array}{l} \text { Density }=\frac{\text { Mass }}{\text { Volume }}=\frac{0.051 \mathrm{~kg}}{75 \times 10^{-6} \mathrm{~m}^{3}}=6.8 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3} \\ \text { sp gr }=\frac{\text { Density of gasoline }}{\text { Density of water }}=\frac{6.8 \times 10^{2} \mathrm{~kg} / \mathrm{m}^{3}}{1000 \mathrm{~kg} / \mathrm{m}^{3}}=0.68 \\ \text { sp gr }=\frac{\text { Mass of } 75 \mathrm{~cm}^{3} \text { gasoline }}{\text { Mass of } 75 \mathrm{~cm}^{3} \text { water }}=\frac{51 \mathrm{~g}}{75 \mathrm{~g}}=0.68 \end{array} $$

Step by step solution

01

Understand the Task

The goal is to find the density and specific gravity of gasoline given its mass and volume. We need to calculate the density using the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \) and then find the specific gravity by comparing it to the density of water, which is \( 1000 \, \text{kg/m}^3 \).

02

Convert Units

First, convert the mass from grams to kilograms: 51 g = 0.051 kg. Next, convert the volume from cm³ to m³ by using the conversion factor \(1 \, \text{m}^3 = 1 \times 10^6 \, \text{cm}^3\), thus \(75 \, \text{cm}^3 = 75 \times 10^{-6} \, \text{m}^3\).

03

Calculate Density

Using the density formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), substitute the mass and volume: Density \(= \frac{0.051 \, \text{kg}}{75 \times 10^{-6} \, \text{m}^3} = 6.8 \times 10^2 \, \text{kg/m}^3 \).

04

Calculate Specific Gravity

Specific gravity (sp gr) is calculated by dividing the density of gasoline by the density of water. Using the formula \( \text{sp gr} = \frac{\text{Density of gasoline}}{\text{Density of water}} \), we have \( \text{sp gr} = \frac{6.8 \times 10^2 \, \text{kg/m}^3}{1000 \, \text{kg/m}^3} = 0.68 \).

05

Verify Specific Gravity from Mass Comparison

Specific gravity can also be calculated by comparing masses for equal volumes. Thus, \( \text{sp gr} = \frac{51 \, \text{g}}{75 \, \text{g}} = 0.68 \) (since \(75 \, \text{g}\) is the mass of the same volume of water).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Unit Conversion

Converting units is an essential step when handling scientific calculations. It ensures that all values are compatible for further processing. In this task, we start with two different unit conversions: mass and volume.

Mass is initially provided in grams (g). To convert grams to kilograms (kg), the standard conversion is applied: there are 1000 grams in a kilogram. Thus, to go from grams to kilograms, simply divide by 1000. Therefore:

• 51 g is converted to 0.051 kg by dividing 51 by 1000.

Volume, on the other hand, is given in cubic centimeters (cm³), but we need it in cubic meters (m³). We know that:

• 1 m³ equals 1,000,000 c³¾Â³.
• Thus, to convert cm³ to m³, multiply by the conversion factor: 75 cm³ becomes 75 x 10^-6 ³¾Â³.

These conversions align the units appropriately for use in the density calculation.

Density Formula

The concept of density reflects how much mass is contained within a specific volume. The density of a substance can be a crucial identifying feature, and is calculated using the formula:

• Density = \(\frac{\text{Mass}}{\text{Volume}}\)

For gasoline, with a mass of 0.051 kg and a volume of 75 x 10^-6 m³, we estimate the density by substituting these values into the formula:

• Density = \(\frac{0.051 \text{ kg}}{75 \times 10^{-6} \text{ m}^3} = 6.8 \times 10^2 \text{ kg/m}^3\).

This result indicates that the mass per cubic meter of gasoline is 680 kg, a useful value when comparing it to other substances.

Specific Gravity Calculation

Specific gravity is a dimensionless quantity. It compares the density of a substance with the density of water at 4°C (which is 1000 kg/m³).

• Specific Gravity (sp gr) = \(\frac{\text{Density of substance}}{\text{Density of water}}\)

By applying this formula to gasoline, we see:

• sp gr = \(\frac{6.8 \times 10^2 \text{ kg/m}^3}{1000 \text{ kg/m}^3} = 0.68\)

This value tells us that gasoline is less dense than water, as it's less than 1. Additionally, you can compare the mass of equivalent water and gasoline volumes. Here:

• Using 75 g of water and gasoline, specific gravity is 0.68 (\(\frac{51 \text{ g}}{75 \text{ g}}\)).

This consistency confirms our density-based calculation.

Mass and Volume Relationship

In understanding the density and specific gravity, it's important to grasp the relationship between mass and volume. In simplest terms:

• Mass reflects the amount of matter in a substance, and it is usually measured in kilograms or grams.
• Volume is the space that matter occupies, measured in units such as cubic meters or cubic centimeters.

This interconnectedness is clearly showcased in the problem. When a given mass occupies a certain volume, it describes how densely packed the substance's molecules are, thus leading to its density value.

For gasoline:

• The given mass is 51 g, while the occupied volume is 75 c³¾Â³. Translating these to kg and m³ helps to utilize the standard international unit system, simplifying scientific communication and computation.

Understanding these basic physical concepts aids in performing accurate scientific calculations and comprehending more complex scientific principles.