Question

A student pours \(44.3 \mathrm{~g}\) of water at \(10^{\circ} \mathrm{C}\) into a beaker containing \(115.2 \mathrm{~g}\) of water at \(10^{\circ} \mathrm{C}\). What are the final mass, temperature, and density of the combined water? The density of water at \(10^{\circ} \mathrm{C}\) is \(1.00 \mathrm{~g} / \mathrm{mL}\).

Short answer

Final mass is 159.5 g, temperature is 10°C, density is 1.00 g/mL.

Step-by-step solution

Calculate the Final Mass

To find the final mass of the combined water, add the masses of the two samples. The first sample has a mass of \(44.3\,\text{g}\) and the second has \(115.2\,\text{g}\). Thus, the total mass is \(44.3 + 115.2 = 159.5\,\text{g}\).

Determine the Final Temperature

Since both samples of water are at the same temperature of \(10^{\circ}\,C\), the final temperature of the combined water remains \(10^{\circ}\,C\).

Calculate the Density

The density of water at \(10^{\circ}\,C\) is given as \(1.00\,\text{g/mL}\). Since the temperature does not change and the substance is composed entirely of water, the density remains \(1.00\,\text{g/mL}\) for the combined mixture.

Key concepts

Mass

Mass is a measure of the amount of matter in an object, typically measured in grams (g) or kilograms (kg). In this exercise, we are combining two samples of water, each with a known mass. To find the final mass of the combined water, we simply add the masses of the individual samples together.
The mass of a substance is straightforward and doesn't depend on its shape or location, such as inside a beaker.
When we added the mass of the first water sample, which is 44.3 g, to the mass of the second sample, 115.2 g, the final mass became 159.5 g.

• The formula to calculate the total mass is:
  \( ext{Final Mass} = ext{Mass}_1 + ext{Mass}_2\)
  Using this approach ensures you account for all the matter in both water samples.
  Remember, to always carry units through the calculation to avoid errors.

So, adding substances together is a nifty way to calculate the total mass, as we have done by simply summing the masses.

Temperature

Temperature is an indicator of the thermal energy in a substance, denoted in degrees Celsius (°C) in this context. It's important to understand that temperature is a property independent of the mass of the object. When dealing with the same substance, like water, and at the same temperature, the final mixture's temperature remains unchanged.
In this exercise, both water samples are initially at 10°C.

• Since no external heat is being added or removed, the final temperature of the combined water remains 10°C.

This happens because of something called thermal equilibrium, where substances in thermal contact reach the same temperature and maintain it without gaining or losing heat.
Therefore, if two samples are mixed and are both at the same initial temperature, the final temperature is the same unless a chemical reaction occurs or heat is transferred in some way.

Water Properties

Water is a unique substance with several interesting properties. One such property is its density, which is defined as mass per unit volume. Here, the density of water at 10°C is given as 1.00 g/mL.
This value means that for every milliliter of water, the mass is 1 gram. This relationship is fundamental, as it allows us to understand and calculate other related concepts, like volume or mass, when one is unknown.

• Even when water is combined, as in this exercise, the density remains constant at this specific temperature unless there is a temperature change or another substance is mixed in.

Water has a high specific heat capacity, meaning it requires a lot of energy to change its temperature, hence maintaining its density under stable conditions.
Understanding how density works with temperature is crucial in many scientific and environmental studies where water is being evaluated.