Question

Write two conversion factors for each of the following. a. a 16\(\%\) (by mass) salt solution b. a density of 1.25 \(\mathrm{g} / \mathrm{mL}\) c. a speed of 25 \(\mathrm{m} / \mathrm{s}\)

Short answer

a. Salt solution conversion factors: \( \frac{16g \; salt}{100g \; solution} \), \( \frac{100g \; solution}{16g \; salt} \) Water conversion factors: \( \frac{84g \; water}{100g \; solution} \), \( \frac{100g \; solution}{84g \; water} \) b. Density conversion factors: \( \frac{1.25g}{1mL} \), \( \frac{1mL}{1.25g} \) c. Speed conversion factors (time-based): \( \frac{25m}{1s} \), \( \frac{1s}{25m} \) Speed conversion factors (minute-based): \( \frac{1500m}{1min} \), \( \frac{1min}{1500m} \)

Step-by-step solution

a. 16% (by mass) salt solution conversion factors

A 16% (by mass) salt solution means 16 grams of salt per 100 grams of solution. We can write two conversion factors based on this information: one considering the mass of salt and the other considering the mass of water. 1. Salt mass conversion factor To find the conversion factor for this, we'll need the relationship between the mass of salt and the total mass of the solution. \(Conversion Factor = \frac{16g \; salt}{100g \; solution}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{100g \; solution}{16g \; salt}\) 2. Water mass conversion factor Since the salt mass is 16 grams and the total mass is 100 grams, the water mass is 84 grams. \(Conversion \; Factor = \frac{84g \; water}{100g \; solution}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{100g \; solution}{84g \; water}\)

b. Density conversion factors (1.25 g/mL)

Given density is 1.25 g/mL, which means 1.25 grams per milliliter. We can write two conversion factors for this: 1. Mass to volume conversion factor \(Conversion \; Factor = \frac{1.25g}{1mL}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{1mL}{1.25g}\) 2. Volume to mass conversion factor (just the inverse of the previous one) \(Conversion \; Factor = \frac{1mL}{1.25g}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{1.25g}{1mL}\)

c. Speed conversion factors (25 m/s)

Given a speed of 25 meters per second, we can write two conversion factors based on time: 1. Conversion factor based on seconds \(Conversion \; Factor = \frac{25m}{1s}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{1s}{25m}\) 2. Conversion factor based on minutes First, we convert 25 meters per second to meters per minute by multiplying by 60 (there are 60 seconds in a minute). \(25 m/s * 60 s/min = 1500 m/min\) \(Conversion \; Factor = \frac{1500m}{1min}\) And the inverse: \(Inverse \; Conversion \; Factor = \frac{1min}{1500m}\)

Key concepts

Mass Percentage

Mass percentage is a way to express the concentration of a component in a solution. It calculates the proportion of a particular substance by comparing its mass to the total mass of the mixture or solution.
Here, a 16% salt solution means that out of every 100 grams of the solution, 16 grams are salt. The remaining 84 grams would be the solvent, typically water. This concept is useful for many applications, including food recipes and chemical formulations.

• It helps in understanding the composition of the solution.
• Easily converted into a fraction for calculations.

For instance, the conversion factor can be written as \( \frac{16g \, \text{salt}}{100g \, \text{solution}} \), meaning there are 16 grams of salt in every 100 grams of total solution. By using the inverse, \( \frac{100g \, \text{solution}}{16g \, \text{salt}} \), we can find the total amount of solution for a given mass of salt.

Density

Density is a measure of how much mass is contained in a given volume. It describes how closely packed the particles are in a material or solution. Typically noted as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³), density is crucial in determining whether an object will float or sink in a fluid.
The given density of 1.25 g/mL provides a relationship: for each milliliter of the solution, there are 1.25 grams of substance.

• Easy identification of a substance's heaviness in a specific volume.
• Crucial for calculations in chemistry and physics.

The conversion factor \( \frac{1.25g}{1mL} \) can be used for finding the mass in grams when the volume is given in milliliters, while \( \frac{1mL}{1.25g} \) aids in the reverse calculation.

Speed Conversion

Speed conversion involves converting the units of speed from one system to another. Speed is generally expressed as the distance covered per unit of time, such as meters per second (m/s) or kilometers per hour (km/h).
For the current example, the speed given is 25 meters per second. This can be expressed directly as \( \frac{25m}{1s} \) or converted into different units like minutes by performing mathematical operations, like \( 25 \, m/s \times 60 \, s/min = 1500 \, m/min \).

• Converts speeds to preferred units according to usage.
• Important for real-world applications like travel and physics.

Using the inverse, \( \frac{1s}{25m} \), converts travel time to distance covered, helping in navigation and timing analyses.

Inverse Conversion

Inverse conversion is a method used to reverse the relation between two units. It allows one to switch between quantities like mass to volume, or speed to time. Inverse factors are essential in chemistry and physics for calculations and unit balancing.
Consider the density of 1.25 g/mL:

• The inverse conversion factor \( \frac{1mL}{1.25g} \) enables us to compute how much volume corresponds to a specific mass.
• Provides clarity in calculations by offering reverse perspectives.

Similarly, for speed, the inverse \( \frac{1s}{25m} \) indicates the fraction of time for every meter traveled. Inverse conversion factors are vital when examining the relationship from a different viewpoint.

Solution Composition

Solution composition refers to the components that make up a solution, which often consist of a solute and a solvent. It can be expressed in terms of concentration like molarity or mass percentage.
A well-understood example is the 16% mass percentage in a salt solution. Knowing the exact composition allows for effective replication and scaling of recipes and reactions.

• Critical in chemical synthesis and industrial processes.
• Helps in predicting solution behavior under different conditions.

In practice, understanding solution composition skills helps in all kinds of formulaic working, whether in a lab or culinary setting, ensuring proper mixing ratios and functionalities.