Question

How many mL of water at \(10^{\circ} \mathrm{C}\) ( 2 significant figures) must be added to \(75 \mathrm{~mL}\) of water at \(35^{\circ} \mathrm{C}\) to obtain a final temperature of \(19^{\circ} \mathrm{C} ?\) (Make the same assumptions as in Question 9.)

Short answer

Answer: About 130 mL of cooler water should be added.

Step-by-step solution

Identify known variables

We are given the following information: - initial volume of warmer water, \(V_{w} = 75 \mathrm{~mL}\) - initial temperature of warmer water, \(T_{w} = 35^{\circ} \mathrm{C}\) - initial temperature of cooler water, \(T_{c} = 10^{\circ} \mathrm{C}\) - final temperature of the mixture, \(T_{f} = 19^{\circ} \mathrm{C}\) Our goal is to find the volume of cooler water \(V_{c}\).

Set up heat balance equation and convert volume to mass

Since the heat gained by the cooler water is equal to the heat lost by the warmer water, we have: $$ m_{c} c (T_{f} - T_{c}) = m_{w} c (T_{w} - T_{f}), $$ where \(m_{c}\) and \(m_{w}\) are the masses of the cooler and warmer water samples, respectively, and \(c\) is the specific heat of water. As the density of water is 1 g/mL, the volumes can be directly converted to masses: \(m_{c} = V_{c}\) and \(m_{w} = V_{w}\) Substitute for masses: $$ V_{c} (T_{f} - T_{c}) = V_{w} (T_{w} - T_{f}) $$

Solve for the unknown volume \(V_{c}\)

Now, plug in the known values and solve for \(V_{c}\): $$ V_{c} (19 - 10) = 75 (35 - 19) $$ $$ V_{c} \times 9 = 75 \times 16 $$ $$ V_{c} = \frac{75 \times 16}{9} $$ Calculate the value of \(V_{c}\): $$ V_{c} \approx 133.3 \mathrm{~mL} $$

Round to 2 significant figures

Finally, round the answer to 2 significant figures: $$ V_{c} \approx 130 \mathrm{~mL} $$ So, about 130 mL of cooler water at \(10^{\circ} \mathrm{C}\) must be added to \(75 \mathrm{~mL}\) of water at \(35^{\circ} \mathrm{C}\) to obtain a final temperature of \(19^{\circ} \mathrm{C}\).

Key concepts

Thermodynamics Basics

Thermodynamics is a branch of physics that deals with heat, work, and temperature, and their relation to energy, radiation, and physical properties of matter. In simple terms, it explains how energy changes within a system and how energy can perform work.

Understanding the principle of energy conservation is crucial in thermodynamics. It states that energy cannot be created or destroyed; it can only be transferred or change forms. A common thermodynamic process is the transfer of heat, which is the energy that is transferred from one system to another as a result of temperature difference.

In our exercise, the thermodynamics concept of a heat balance was used to determine the necessary volume of water to achieve a desired final temperature. This involved looking at the energy moving between two bodies of water at different temperatures and understanding how they would interact and eventually reach thermal equilibrium.

Understanding Specific Heat

Specific heat is a measure of how much heat energy is needed to raise the temperature of a substance by a degree Celsius (or Kelvin). It's a property that depends on the material and is denoted by the symbol c. In the International System of Units (SI), it's measured in joules per kilogram per degree Celsius (J/kg°C).

The specific heat of water, for instance, is relatively high at approximately 4.18 J/g°C, which means it takes a considerable amount of energy to raise its temperature. This high specific heat capacity is why water is used as a coolant in various applications and contributes to the moderation of Earth's climate.

In the provided exercise, we exploit the fact that water's specific heat is constant to set up the heat balance equation between the warmer and cooler water, allowing us to find the unknown volume.

Mixtures Temperature Calculation

Calculating the final temperature of a mixture involves understanding the heat transfer between the constituents of the mixture. When two bodies at different temperatures are mixed, heat will flow from the hotter object to the colder one until equilibrium is reached.

In the context of the exercise, we combined two bodies of water at different temperatures and sought the final uniform temperature after they interact. To do this, we created a heat balance equation based on the principle that the heat lost by the warmer water will equal the heat gained by the cooler water. This is a reflection of energy conservation in a closed system, where all the heat released by the warmer body must be absorbed by the colder one, assuming no heat is lost to the surroundings.

The temperature change for both components in the mixture is directly proportional to their masses and the specific heat of water, which allows us to set up an equation to solve for the unknown quantity—either the mass (or, equivalently for water, the volume) of one of the components or the final temperature of the mixture.