Question

Sodium chloride is added in cooking to enhance the flavor of food. When \(10.00 \mathrm{~g}\) of \(\mathrm{NaCl}\) are dissolved in \(200.0 \mathrm{~mL}\) of water at \(25.0^{\circ} \mathrm{C}\) in a coffee-cup calorimeter, \(669 \mathrm{~J}\) of heat are absorbed. (You can make the following assumptions about the solution: volume \(=200.0 \mathrm{~mL},\) density \(=1.00 \mathrm{~g} / \mathrm{mL}\), specific heat \(\left.=4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\) (a) Is the solution process exothermic? (b) What is \(q_{\mathrm{H}_{2} \mathrm{O}}\) ? (c) What is the final temperature of the solution?

Short answer

What is the value of \(q_{\mathrm{H}_{2} \mathrm{O}}\), and what is the final temperature of the solution? Answer: The dissolving process is endothermic. The value of \(q_{\mathrm{H}_{2} \mathrm{O}}\) is 669 J, and the final temperature of the solution is approximately \(25.8^{\circ}\mathrm{C}\).

Step-by-step solution

(a) Identifying if the solution process is exothermic or endothermic

When a process releases heat, it is called an exothermic process. Conversely, if a process absorbs heat, it is called an endothermic process. In this case, the process absorbs 669 J of heat, so it is an endothermic process.

(b) Calculating \(q_{\mathrm{H}_{2} \mathrm{O}}\) (heat absorbed or released by water)

Since the process is endothermic, the heat absorbed by the NaCl solution will be the heat released by water. Hence the value of \(q_{\mathrm{H}_{2}\mathrm{O}}\) can be found using the formula: \(q_{\mathrm{H}_{2}\mathrm{O}} = -q_{\mathrm{NaCl}}\) \(q_{\mathrm{H}_{2}\mathrm{O}} = -(-669 \mathrm{~J})\) \(q_{\mathrm{H}_{2}\mathrm{O} = 669 \mathrm{~J}\)

(c) Calculating the final temperature of the solution

To find the final temperature of the solution, we will use the heat transfer equation: \(q_{\mathrm{H}_{2} \mathrm{O}}= m_{\mathrm{H}_{2} \mathrm{O}}\times c_{\mathrm{H}_{2} \mathrm{O}}\times \Delta T\) Here, \(q_{\mathrm{H}_{2} \mathrm{O}}\) is the heat absorbed or released by water (which we found in step b), \(m_{\mathrm{H}_{2} \mathrm{O}}\) is the mass of water (found by multiplying the density by the volume), \(c_{\mathrm{H}_{2} \mathrm{O}}\) is the specific heat of water, and \(\Delta T\) is the change in temperature. First, we will find the mass of water: \(m_{\mathrm{H}_{2} \mathrm{O}} = \rho V = 1.00 \mathrm{~g/mL} \times 200.0 \mathrm{~mL} = 200.0 \mathrm{~g}\) Now, we will plug in the values and solve for \(\Delta T\): \(669 \mathrm{~J} = 200.0\mathrm{~g} \times 4.18 \mathrm{~J/g\cdot^{\circ}C} \times \Delta T\) Dividing both sides by \(200.0\mathrm{~g} \times 4.18 \mathrm{~J/g\cdot^{\circ}C}\), we get: \(\Delta T = \frac{669\mathrm{~J}}{200.0\mathrm{~g} \times 4.18 \mathrm{~J/g\cdot^{\circ}C}} \approx 0.80^{\circ}\mathrm{C}\) Finally, we will find the final temperature: \(T_{final} = T_{initial} + \Delta T = 25.0^{\circ}\mathrm{C} + 0.80^{\circ}\mathrm{C} \approx 25.8^{\circ}\mathrm{C}\) So, the final temperature of the solution is approximately \(25.8^{\circ}\mathrm{C}\).

Key concepts

Endothermic Process

To comprehend thermochemistry deeply, it's essential to grasp the concept of an endothermic process. This term refers to a chemical reaction or physical change that absorbs energy, typically in the form of heat, from its surroundings. Unlike its counterpart, the exothermic process which releases heat, an endothermic process requires energy input to proceed. This is often observed as a drop in temperature in the surrounding environment, as heat is drawn into the system to fuel the reaction.

In the exercise we are discussing, the dissolving of sodium chloride (table salt) in water exemplifies an endothermic process, where the system (salt solution) requires energy to break the ionic bonds of the salt. This energy is taken from the water in the calorimeter, thereby causing the water to cool down as it gives up heat energy. Remember, recognizing the energy flow is the key to distinguishing between endothermic and exothermic processes in chemical reactions.

Calorimetry

Unveiling the heat dynamics in physical and chemical processes calls for an understanding of calorimetry. Calorimetry is the science of measuring the amount of heat transferred to or from an object, which is crucial for understanding thermal energy changes in a system. The device used for these measurements is known as a calorimeter.

In our textbook problem, a coffee-cup calorimeter is used to measure the heat involved in the dissolution of sodium chloride in water. The calorimeter is a simple insulated container that helps in determining the heat absorbed by the process. During the experiment, any heat absorbed or released by the content of the calorimeter is assumed not to be lost to the surroundings due to insulation. Through calorimetry, we can investigate the energy changes involved and quantify them, as shown by the heat absorbed (\( 669 \text{ J} \)) by the NaCl solution in the experiment.

Heat Transfer Equation

Moving further into thermodynamic investigations, we encounter the heat transfer equation, a mathematical framework allowing us to quantify the thermal energy change within a system. It is commonly stated as:

\( q = mc\Delta T \),
where \( q \) is the heat transferred, \( m \) is the mass, \( c \) is the specific heat capacity, and \( \Delta T \) is the change in temperature. This equation is pivotal in calculating the energy involved in processes where temperature change occurs, like heating or cooling substances.

In the context of our example, we applied the heat transfer equation to solve for the final temperature of the water after absorbing the heat from the salt solution. The specific heat capacity (\( c \) value) and the mass of the water enabled us to capitalize on the given information, showing that the final temperature of the solution is slightly higher due to the endothermic nature of the process. The focus on these calculations is vital as they apply to various scientific and engineering problems, from designing heating systems to understanding the thermal balance in environmental sciences.