Question

The density of water at \(40^{\circ} \mathrm{C}\) is \(0.992 \mathrm{~g} / \mathrm{mL}\). What is the volume of \(27.0 \mathrm{~g}\) of water at this temperature?

Short answer

The volume is approximately 27.220 mL.

Step-by-step solution

Understanding the relationship between mass, volume, and density

Density is defined as mass per unit volume, which can be expressed with the formula \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). The given information tells us the density of water at 40°C is \(0.992 \ \text{g/mL}\), and we are to find the volume of 27.0 grams of water.

Rearranging the formula to solve for volume

We want to find the volume, so we need to rearrange the density formula to solve for it. Rearranging the formula gives: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \).

Substituting the known values into the formula

Now, substitute the known values into the formula: \( \text{Volume} = \frac{27.0 \ \text{g}}{0.992 \ \text{g/mL}} \).

Calculating the volume

Carry out the division: \( \text{Volume} = \frac{27.0}{0.992} \approx 27.220 \ \text{mL} \).

Concluding the solution

The calculation shows that the volume of 27.0 grams of water at 40°C is approximately 27.220 mL, due to the given density of 0.992 g/mL.

Key concepts

Mass and Volume Relationship

Understanding the relationship between mass and volume is fundamental when working with concepts like density. Mass is the amount of matter in an object and is typically measured in units like grams (g) or kilograms (kg). Volume, on the other hand, describes the amount of space that an object occupies and is typically measured in units like milliliters (mL) or liters (L).
Linking mass and volume together, density allows us to understand how much matter is packed into a given volume of space. A denser substance will have more mass in the same volume compared to a less dense substance. In the context of our exercise, water's density at 40°C helps us relate its mass to its volume, enabling precise calculations.

Density Formula

The density formula is a key tool in connecting mass and volume. It is presented mathematically as \( \text{Density} = \frac{\text{Mass}}{\text{Volume}} \). This formula tells us how much mass of a substance is contained in a given volume.
If we have the values for both mass and volume, we can calculate the density directly. Conversely, if we know the density and either the mass or volume, we can rearrange the formula to find the missing quantity. This flexibility makes the density formula versatile when solving various kinds of problems involving mass and volume.
In our specific problem, knowing the density of water at 40°C and its mass allows us to effectively determine the water's volume.

Volume Calculation

To find the volume when we have mass and density, we simply need to rearrange the density formula: \( \text{Volume} = \frac{\text{Mass}}{\text{Density}} \). This gives us a straightforward way to calculate the space that a given amount of substance occupies under specific conditions.
Using the values provided in the exercise, we determined the volume of 27.0 grams of water by substituting the mass (27.0 g) and the density of water at 40°C (0.992 g/mL) into the equation. The calculation \( \text{Volume} = \frac{27.0}{0.992} \approx 27.220 \ \text{mL} \) gives us the result. Thus, at 40°C, 27.0 grams of water occupies approximately 27.220 mL.

Density of Water at 40°C

The density of water changes with temperature, which is important when performing calculations involving physical properties. At 40°C, the density of water is approximately 0.992 g/mL. This specific value is crucial for accurately determining the volume of water or related calculations at this temperature.
This temperature variation is because as water heats, the molecules move more vigorously and spread slightly apart, lowering the density.
Having the correct density value is essential for the accuracy of any calculations, such as the one performed in our exercise, ensuring that we consider the specific characteristics of water at different temperatures.