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Chapter 1: Problem 40

A \(5-\mathrm{m}^{3}\) container is filled with \(900 \mathrm{~kg}\) of granite (density of \(2400 \mathrm{~kg} / \mathrm{m}^{3}\) ). The rest of the volume is air, with density equal to \(1.15 \mathrm{~kg} / \mathrm{m}^{3}\). Find the mass of air and the overall (average) specific volume.

Short Answer

Expert verified
Mass of air is approximately 5.32 kg. The overall specific volume is about 0.00552 m鲁/kg.

Step by step solution

01

Calculate the Volume Occupied by Granite

The density of granite is given as \(2400 \, \text{kg/m}^3\). By using the formula for density, \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \), we can find the volume of the granite. Rearrange the formula to solve for volume: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). Substitute the given values: \( \text{Volume of Granite} = \frac{900 \, \text{kg}}{2400 \, \text{kg/m}^3} = 0.375 \, \text{m}^3 \).
02

Calculate the Volume Occupied by Air

The total volume of the container is \(5 \, \text{m}^3\). Since part of this is occupied by granite, subtract the volume of granite (calculated in Step 1) from the total volume to find the volume occupied by air: \( \text{Volume of Air} = 5 \, \text{m}^3 - 0.375 \, \text{m}^3 = 4.625 \, \text{m}^3 \).
03

Calculate the Mass of Air

Using the density of air given as \(1.15 \, \text{kg/m}^3\), calculate the mass of the air using the formula \( \text{Mass} = \text{Density} \times \text{Volume} \). Substitute the values we found: \( \text{Mass of Air} = 1.15 \, \text{kg/m}^3 \times 4.625 \, \text{m}^3 = 5.31875 \, \text{kg} \).
04

Calculate the Overall Specific Volume

The total mass in the container is the sum of the mass of granite and the mass of air, which is \(900 \, \text{kg} + 5.31875 \, \text{kg} = 905.31875 \, \text{kg} \). The specific volume \(u\) is given by \(u = \frac{\text{Total Volume}}{\text{Total Mass}}\). Substitute the values to get \(u = \frac{5 \, \text{m}^3}{905.31875 \, \text{kg}} \approx 0.00552 \, \text{m}^3/\text{kg}\).

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Key Concepts

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

Density Calculation
Density is a fundamental property of matter that describes the mass per unit volume of a substance. It offers a way to determine how compact a material is within a defined space.
To calculate density, we use the formula:
  • Density, \( \rho = \frac{\text{Mass}}{\text{Volume}} \)
In our example of granite, knowing the mass (900 kg) and density (2400 kg/m鲁), we can rearrange the formula to solve for the volume occupied by the granite:
  • \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \)
  • \( \text{Volume of Granite} = \frac{900 \text{ kg}}{2400 \text{ kg/m}^3} = 0.375 \text{ m}^3 \)
Density calculations can be applied to find the properties of different parts of a mixture, as seen when determining how much of the container is filled with granite versus air.
Specific Volume
Specific volume is a useful concept in thermodynamics, representing the volume occupied per unit mass of a substance. It is essentially the inverse of density and is often denoted by the letter \( v \).
The formula to calculate specific volume is:
  • \( v = \frac{\text{Volume}}{\text{Mass}} \)
In this context, finding the overall specific volume means considering the entire container's volume and mass, which includes both the granite and the air. The total volume of the container is known (5 m鲁), and the total mass is the sum of the mass of granite and the mass of air:
  • Total Mass = 900 kg (granite) + 5.31875 kg (air) = 905.31875 kg
Then, we calculate the specific volume:
  • \( v = \frac{5 \text{ m}^3}{905.31875 \text{ kg}} \approx 0.00552 \text{ m}^3/\text{kg} \)
Specific volume helps to understand how dispersed the mass of the substance is within the given volume.
Mass Calculation
To determine the mass of a substance when its density and volume are known, we use the following relationship:
  • Mass = Density \( \times \) Volume
This can help identify how much a specific material weighs within a particular space.In the problem, we calculated the mass of the air filling the remaining space in the container after the granite:
  • Volume of Air = Total Volume - Volume of Granite = 5 m鲁 - 0.375 m鲁 = 4.625 m鲁
  • Density of Air = 1.15 kg/m鲁
  • Mass of Air = 1.15 kg/m鲁 \( \times \) 4.625 m鲁 = 5.31875 kg
Mass calculation is crucial for determining how different materials contribute to the overall weight within a space.
Volume Calculation
Calculating volume involves understanding how much space an object or substance occupies, typically described in cubic meters (m鲁) or liters. The relationship between mass, density, and volume is given by the equation:
  • Volume = \( \frac{\text{Mass}}{\text{Density}} \)
Volume calculation becomes particularly useful for differentiating between various components in a mixture, such as in our example where we distinguish between granite and air.From the problem, we calculated the volume occupied by granite first, and then by air:
  • Using the formula, we find the Volume of Granite = \( \frac{900 \text{ kg}}{2400 \text{ kg/m}^3} = 0.375 \text{ m}^3 \)
  • Once we know how much space granite occupies, the remaining volume of the container is the space taken up by air: Volume of Air = 5 m鲁 - 0.375 m鲁 = 4.625 m鲁
Volume calculation is an essential step in problems involving mixtures or systems with multiple components.

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