Problem 97 - Physics

Question

A gold prospector finds a solid rock that is composed solely of quartz and gold. (a) How is the total mass \(m_{\mathrm{T}}\) of the rock related to the mass \(m_{\mathrm{G}}\) of the gold and the mass \(m_{\mathrm{Q}}\) of the quartz? (b) What is the relationship between the total volume \(V_{\mathrm{T}}\) of the rock, the volume \(V_{\mathrm{G}}\) of the gold, and the volume \(V_{\mathrm{Q}}\) of the quartz? (c) How is the volume of a substance (gold or quartz) related to the mass of the substance and its density? The mass and volume of the rock are, respectively, \(12.0 \mathrm{~kg}\) and \(4.00 \times 10^{-3} \mathrm{~m}^{3}\). Find the mass of the gold in the rock.

Step-by-Step Solution

Answer

(a) Total mass is the sum of gold and quartz masses: \(m_{\mathrm{T}} = m_{\mathrm{G}} + m_{\mathrm{Q}}\). (b) Total volume is the sum of gold and quartz volumes: \(V_{\mathrm{T}} = V_{\mathrm{G}} + V_{\mathrm{Q}}\). (c) Volume is mass divided by density: \(V = \frac{m}{\rho}\).

Step 1: Understand Total Mass Relationship

The total mass of the rock, \(m_{\mathrm{T}}\), is the sum of the mass of the gold, \(m_{\mathrm{G}}\), and the mass of the quartz, \(m_{\mathrm{Q}}\). So, we can express this relationship as: \[ m_{\mathrm{T}} = m_{\mathrm{G}} + m_{\mathrm{Q}}. \]

Step 2: Understand Total Volume Relationship

Since volume is an additive property for non-reactive materials, the total volume of the rock, \(V_{\mathrm{T}}\), is the sum of the volume of the gold, \(V_{\mathrm{G}}\), and the volume of the quartz, \(V_{\mathrm{Q}}\). Thus, the relationship is: \[ V_{\mathrm{T}} = V_{\mathrm{G}} + V_{\mathrm{Q}}. \]

Step 3: Relating Volume, Mass, and Density

The volume \(V\) of a substance can be found using its mass \(m\) and density \(\rho\). The relationship is given by: \[ V = \frac{m}{\rho}, \] where \(\rho\) is the density of the substance. Therefore, for a given mass, you can compute the volume using this formula.

Key Concepts

Mass of Substances

In understanding the mass of substances, it is essential to recognize that the total mass of a mixture, like a rock composed of gold and quartz, is the sum of the masses of its individual components. The mass of each substance contributes to the total mass. For example, if you have a rock containing both gold and quartz, the masses of the gold and quartz are simply added together to find the total mass of the rock.

This can be expressed mathematically as:

• Total Mass Formula: The total mass of the rock, \( m_{\mathrm{T}} \), equals the mass of the gold, \( m_{\mathrm{G}} \), plus the mass of the quartz, \( m_{\mathrm{Q}} \). Therefore, \( m_{\mathrm{T}} = m_{\mathrm{G}} + m_{\mathrm{Q}} \).

Recognizing this relationship helps in determining how much of each substance is present in a composite material. As long as the substances do not chemically react to form a new one, this additive property holds true.

Volume of Substances

Volume, much like mass, can also be considered an additive property for non-reactive substances. When dealing with a rock composed of gold and quartz, the total volume is the sum of the individual volumes of the gold and quartz within the rock. This is particularly straightforward as gold and quartz in their solid states do not react with each other.

The relationship can be stated as follows:

• Total Volume Formula: The total volume of the rock, \( V_{\mathrm{T}} \), is the sum of the volume of the gold, \( V_{\mathrm{G}} \), and the quartz, \( V_{\mathrm{Q}} \); hence, \( V_{\mathrm{T}} = V_{\mathrm{G}} + V_{\mathrm{Q}} \).

Understanding this makes it easy to calculate how much space each substance occupies. This is especially handy in geology and material sciences where the volume calculations help in determining the density and other properties of composite materials.

Density Formula

The concept of density helps us relate the mass and volume of a substance. Density is defined as the mass of a substance per unit volume, and it determines how much mass is contained in a given volume of a material. The formula for density is expressed as:\( \rho = \frac{m}{V} \), where \( \rho \) represents density, \( m \) is mass, and \( V \) is volume.

This relationship can be rearranged to find volume if mass and density are known, using the formula:

• Volume from Mass and Density: \( V = \frac{m}{\rho} \). This tells us how to calculate the volume of a substance if we know its mass and density.

Recognizing this relationship is key, especially when dealing with materials of different densities in a composite substance. By using this formula, you can solve for any of the three variables (mass, volume, or density) if the other two are provided. This plays a crucial role in fields like chemistry and physics, aiding in the understanding of material properties and helping us make informed calculations about a substance's structure.