Question

If ethanol (grain alcohol) has a density of \(0.785 \mathrm{g} / \mathrm{mL}\) calculate the volume of \(82.5 \mathrm{g}\) of ethanol.

Short answer

The volume of 82.5 g of ethanol is approximately 105.1 mL.

Step-by-step solution

Write down the given values

We have the following values given in the problem: Density of ethanol (D) = 0.785 g/mL Mass of ethanol (M) = 82.5 g Our goal is to find the volume of ethanol (V).

Write the formula to find the volume

The formula to find the volume with density and mass is: Volume (V) = Mass (M) / Density (D)

Substitute the given values into the formula

Now, substitute the given values of mass and density into the formula for volume: V = 82.5 g / 0.785 g/mL

Calculate the volume

Now, simply divide the values: V = \( \frac{82.5}{0.785} \) mL ≈ 105.1 mL Therefore, the volume of 82.5 g of ethanol is approximately 105.1 mL.

Key concepts

Volume Calculation

Volume calculation is a fundamental concept in chemistry and physics that involves determining the amount of space that a substance occupies. It's essential for understanding how to measure and compare different substances. In the context of the exercise, to find the volume of ethanol, we used the formula:\[ V = \frac{M}{D} \]where \( V \) is the volume, \( M \) is the mass, and \( D \) is the density of the substance. This formula comes in handy when you have the mass and density and need to calculate the volume.

• Mass - The amount of matter in the substance, usually measured in grams (g).
• Density - The mass per unit volume, expressed in grams per milliliter (g/mL).
• Volume - The amount of space occupied by the substance, measured in milliliters (mL).

Applying the values from the problem, we substitute into the formula:\[ V = \frac{82.5 \text{ g}}{0.785 \text{ g/mL}} \]Calculating gives approximately 105.1 mL. Thus, the ethanol occupies 105.1 mL of space. This calculation is crucial when you need to measure liquids accurately for experiments.

Mass and Density Relationship

Understanding the relationship between mass and density is key for solving many practical problems in science. Mass is an invariant measure of matter, while density describes how tightly that matter is packed into a space.

• Density Formula: The relationship is defined as: \[ D = \frac{M}{V} \] where density (\( D \)) is mass (\( M \)) divided by volume (\( V \)).
• This makes density a derived quantity based on mass and volume.
• Knowing two of these quantities lets you calculate the third.

In our problem, we have the density and mass of ethanol; hence, we calculate the volume. This illustrates their interconnectedness:- If the density is increased while mass remains constant, volume will decrease.- If mass increases without changing density, volume must also increase.By manipulating the relationship properly, you can solve for any missing variable with confidence. Therefore, mastering this relationship assists in resolving not just textbook problems but real-world scenarios as well.

Unit Conversion

Unit conversion involves changing the measurement of a property from one unit to another. While the example didn't involve conversion, it’s essential to understand this skill. In density-related calculations, matching units correctly ensures the formulas give accurate results.

• Base Units: The mass is often in grams (g), the volume in milliliters (mL), aligning well with density units, \( \text{g/mL} \).
• Conversion Between Units - Common conversions include:
  • 1 liter = 1000 milliliters
  • 1 kilogram = 1000 grams
• Keeping the units consistent during calculations avoids errors.

For example, if you have a density stated in \( \text{kg/L} \), converting it into \( \text{g/mL} \) might be required if your mass is in grams and volume in milliliters. To do this, remember:- Multiply or divide using the appropriate conversion factor, ensuring all measurements align.This skill is invaluable as your calculations become more complex or varied, accommodating a wide range of scientific and real-world tasks.