Question

Silver has a density of \(10.5 \mathrm{~g} \mathrm{~cm}^{-3}\). Express this as an equivalence between mass and volume for silver. Write two conversion factors that can be formed from this equivalence for use in calculations.

Short answer

The equivalence is 1 cm^3 of silver equals 10.5 g. Conversion factors are \(10.5 \mathrm{~g/cm^3}\) for volume to mass and \(1/10.5 \mathrm{~cm^3/g}\) for mass to volume.

Step-by-step solution

Understanding Density

Density is defined as the mass of a substance divided by its volume. It is expressed as a ratio where the numerator is mass and the denominator is volume. For silver, the density is given as \(10.5 \mathrm{~g/cm^3}\).

Writing the Equivalence

Since density is mass per unit volume, the equivalence can be written as: '1 cm^3 of silver has a mass of 10.5 g'. This directly states the relationship between the mass of silver and its volume. The equivalence is formed by setting the density equal to the mass divided by the volume.

Formulating Conversion Factors

Using the equivalence, two conversion factors can be formed. For conversion from volume to mass: \(10.5 \mathrm{~g/cm^3}\) (1 cm^3 of silver is equivalent to 10.5 g of silver). For conversion from mass to volume: \(1/10.5 \mathrm{~cm^3/g}\) (1 g of silver is equivalent to \(1/10.5\) cm^3 of silver).

Key concepts

Density Calculations

Density is a fundamental concept in physical sciences that relates the mass of a substance to its volume. In essence, it answers the question: 'How much mass is packed into a given volume?' Understanding this concept is crucial for a range of applications, from designing ships to measuring the purity of precious metals like silver.

To calculate density, you take the mass (usually in grams) and divide it by the volume (commonly in cubic centimeters). Mathematically, it's expressed as the formula: \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). In the case of silver, its density is given to be \(10.5 \text{ g/cm}^3\), which is a relatively high density indicative of its status as a precious metal. Knowing how to compute and interpret density paves the way for more advanced concepts like buoyancy, and is also essential in stoichiometry problems in chemistry where mass-to-volume conversions are necessary.

Mass-Volume Relationship

The relationship between mass and volume is at the very heart of the concept of density. With silver's density at \(10.5 \text{ g/cm}^3\), you can infer that for every cubic centimeter of silver, the mass is 10.5 grams. This direct proportionality forms the basis for many mathematical models and solutions in chemistry and physics.

Having a clear understanding of this relationship allows students to solve practical problems. For example, if a jeweler needs to know the mass of a piece of silver jewelry, they can simply calculate the volume of the piece and use the density to find the mass. Conversely, knowing the mass, they could determine the volume.

This interplay between mass and volume also factors into chemical equations and reactions where precise measurements are critical for accuracy. Therefore, a solid grasp of how mass and volume correlate is indispensable for both theoretical understanding and practical problem-solving.

Chemistry Problem Solving

Problem-solving in chemistry often involves navigating complex relationships between variables, and it requires a systematic approach to dissect problems into manageable parts. In the context of the mass-volume relationship, the density of a substance can serve as a bridge that links these two variables. By forming an equivalence as demonstrated with silver, you can extrapolate either value when provided with the other.

For instance, if you know the mass of silver and need to find the volume, you utilize the formula derived from the given density. Just as in the equation \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). This process underscores the importance of understanding the underlying principles that govern chemical behavior and properties.

Moreover, adopting units consistent with the extensive use in chemistry, such as grams for mass and cubic centimeters for volume, simplifies the calculations and avoids potential errors due to unit conversion. Emphasizing unit analysis and thoughtfully arranging variables maintains accuracy and fosters proficient problem-solving skills in chemistry.

Conversion Factors in Chemistry

Conversion factors are essential tools in chemistry that assist with the translation of one unit of measure to another. When dealing with densities and the mass-volume relationship, these factors enable the seamless interpolation between mass and volume. Based on the silver density example, the two most straightforward conversion factors would be \(10.5 \text{ g/cm}^3\) and its reciprocal \(1/10.5 \text{ cm}^3/g\).

These allow for swift conversions in calculations without needing to perform complex algebra each time. For practical problem-solving, having this sort of 'shorthand' can streamline the process and minimize error. When moving between mass and volume, it's recommended to write out the conversion factors explicitly at the start of a problem. This clarifies the pathway of the calculation and serves as a double-check to ensure the accuracy of dimensional analysis.

By consistently practicing with conversion factors, students can develop a strong intuitive sense of how to convert units across various chemical contexts, enhancing both their efficiency and effectiveness in tackling chemistry problems.