Question

The density of a certain liquid is \(1.15 \mathrm{~g} / \mathrm{mL}\). What mass in grams of the liquid is needed to fill a \(50.00\) -mL container? Do this problem by the method of algebraic manipulation, beginning with the equation density \(=\) mass/volume and showing all steps.

Short answer

The mass of the liquid needed to fill the 50.00 mL container is \(57.50\mathrm{g}\).

Step-by-step solution

Write the equation for density

As we know, the formula for density is given by: density = mass/volume

Plug in the given values

We are given the values of density and volume. The density is 1.15 g/mL, and the volume is 50.00 mL. Now, plug these values into the density formula: \(1.15 \frac{\mathrm{g}}{\mathrm{mL}} = \frac{\mathrm{mass}}{50.00 \mathrm{mL}}\)

Solve for mass

To find the mass, we need to isolate the mass on one side of the equation. We will do this by multiplying both sides by the volume, which is 50.00 mL: \(1.15 \frac{\mathrm{g}}{\mathrm{mL}} * 50.00 \mathrm{mL} = \mathrm{mass}\)

Simplify the equation

Now, we can simplify the equation and remove any unnecessary units: \((1.15\mathrm{g}) * (50.00 /\cancel{\mathrm{mL}}) * (\cancel{\mathrm{mL}}) = \mathrm{mass}\)

Calculate the mass

Our equation is now in its simplest form, and we can simply multiply 1.15 g by 50.00 to find the mass. \((1.15\mathrm{g}) * (50.00) = 57.50\mathrm{g}\) So, the mass of the liquid needed to fill the 50.00 mL container is 57.50 grams.

Key concepts

Understanding Mass and Volume

To effectively solve density problems, it's crucial to grasp the fundamental concepts of mass and volume. Mass refers to the amount of matter in an object, typically measured in grams (g).
Volume, on the other hand, is the space that an object occupies. In this exercise, the volume is given as 50.00 mL. The relationship between mass and volume is essential in density calculations. Mass does not change regardless of location, whether on Earth or anywhere else. Volume, too, is constant for the given conditions of this problem.
Understanding these two helps us work through the density formula correctly and solve for mass or volume when other values are given.

Algebraic Manipulation in Density Calculations

Algebraic manipulation involves rearranging equations to isolate the desired variable. This skill is especially valuable in science and chemistry.
For instance, in our density formula \[ \text{density} = \frac{\text{mass}}{\text{volume}} \]we start by identifying the variable we need to solve for. Here, it's the mass.- To isolate mass, we multiply both sides of the equation by volume: \[ \text{density} \times \text{volume} = \text{mass} \]- This straightforward manipulation clears the fraction, allowing us to determine mass directly by substituting known values.Using algebra, we transformed the problem to find the mass, showing how basic algebraic operations can solve real-world science and math problems efficiently here.

Chemical Problem Solving Made Simple

Chemical problem solving can seem daunting at first, but breaking it into smaller steps makes it manageable. In this problem, our focus is on determining the mass of a liquid using known density and volume.
Here's how you can approach similar problems: - Start with the known formula, such as the density formula in this case. - Identify your given values (density and volume) and what you need to find (mass). - Use algebraic manipulation to rearrange the equation and solve for the unknown variable. This systematic approach ensures you follow a clear path from start to finish, avoiding common mistakes. Chemical problem solving is often about understanding relationships and applying them correctly.
Each step in the problem-solving process builds on the last, leading you to the final answer efficiently.

The Density Formula Explained

At the heart of this problem is the density formula, a fundamental concept in chemistry and physics. Density is defined as mass per unit volume and is usually expressed as grams per milliliter (g/mL) or kilograms per cubic meter (kg/m³).- The formula is: \[ \text{density} = \frac{\text{mass}}{\text{volume}} \]- Using this formula, you can solve for mass if you know density and volume: \[ \text{mass} = \text{density} \times \text{volume} \] Density tells us how much space a substance occupies given a certain mass. It also helps to identify substances and understand their properties.
By manipulating this formula, you can determine one variable if the other two are known, as we did in this exercise using given density and volume to find the mass.
The density formula is versatile, proving useful across various scientific fields for research, problem-solving, and material characterization.