Question

Urea, \(\left(\mathrm{NH}_{2}\right)_{2} \mathrm{CO}\), is used in the manufacture of resins and glues. When \(5.00 \mathrm{~g}\) of urea is dissolved in \(250.0 \mathrm{~mL}\) of water \((d=1.00 \mathrm{~g} / \mathrm{mL})\) at \(30.0^{\circ} \mathrm{C}\) in a coffee-cup calorimeter, \(27.6 \mathrm{~kJ}\) of heat is absorbed. (a) Is the solution process exothermic? (b) What is \(q_{\mathrm{H}_{2} \mathrm{O}}\) ? (c) What is the final temperature of the solution? (Specific heat of water is \(4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\).) (d) What are the initial and final temperatures in \({ }^{\circ} \mathrm{F}\) ?

Short answer

What are the initial and final temperatures of the solution in °C and °F? Answer: The solution process is endothermic as it absorbs heat. The heat absorbed by water during the dissolution is 27,600 J. The initial temperature is 30.0°C (86.0°F) and the final temperature is 56.4°C (133.5°F).

Step-by-step solution

(a) Identifying the process as exothermic or endothermic

Since the heat is absorbed (instead of being released) during the solution process, the process is endothermic (not exothermic).

(b) Finding the heat absorbed by water, \(q_{H_2O}\)

Since the process is endothermic and absorbs 27.6 kJ (or 27600 J) of heat, the heat absorbed by the water during the solution process can be represented as: \(q_{H_2O} = 27600 \ \text{J}\).

(c) Calculating the final temperature of the solution

To find the final temperature of the solution, we can use the formula for heat transfer: \(q = mcΔT\), where \(q\) is the heat absorbed or released, \(m\) is the mass of the substance, \(c\) is the specific heat, and \(ΔT\) is the change in temperature. In this case, the mass of water (\(m\)) is 250.0 g (since the density of water is 1.00 g/mL), the initial temperature (\(T_i\)) is 30.0°C, and the specific heat of water (\(c\)) is 4.18 J/g·°C. We can rearrange the formula to solve for the change in temperature (\(ΔT\)): \(ΔT = \frac{q}{mc} = \frac{27600 \ \text{J}}{250.0 \ \text{g} \cdot 4.18 \ \text{J/g} \cdot{ }^{\circ} \mathrm{C}} = \frac{27600}{1045} = 26.4{ }^{\circ} \mathrm{C}\) Now we can find the final temperature (\(T_f\)) by adding the change in temperature to the initial temperature: \(T_f = T_i + ΔT = 30.0{ }^{\circ} \mathrm{C} + 26.4{ }^{\circ} \mathrm{C} = 56.4{ }^{\circ} \mathrm{C}\)

(d) Converting the initial and final temperatures to °F

To convert the initial and final temperatures from °C to °F, we can use the formula: \(T_{(°F)} = T_{(°C)} \cdot \frac{9}{5} + 32\) For the initial temperature: \(T_i{(}^\circ F{)} = 30.0{ }^\circ C \cdot \frac{9}{5} + 32 = 54{ }^\circ F\) For the final temperature: \(T_f{(}^{\circ}F{)} = 56.4{ }^{\circ} \mathrm{C} \cdot \frac{9}{5} + 32 = 133.5{ }^{\circ} \mathrm{F}\)

Key concepts

Endothermic Process

An endothermic process is one in which the system absorbs heat from its surroundings. Imagine you're dissolving urea in water and the solution feels cold—it means the system took in heat. This absorption of heat is a hallmark of an endothermic process.
The energy required is absorbed rather than released, as is typical in exothermic processes. In the context of the provided exercise, since 27.6 kJ of heat was absorbed when dissolving urea, we can confidently say this is an endothermic process.
Here's the takeaway:

• If the heat flows into a system, it's endothermic.
• The surroundings will lose heat as the system absorbs it.

Understanding this helps us classify chemical processes and predict how systems will behave in various conditions.

Heat Transfer Calculations

In thermodynamics, calculating heat transfer involves understanding how heat energy moves through substances. To determine the heat absorbed or released, we utilize the formula: \[ q = mc\Delta T \] where \(q\) is the heat transferred, \(m\) is the mass of the substance, \(c\) is the specific heat capacity, and \(\Delta T\) is the temperature change.
In the example of our urea solution, the heat absorbed by the water was 27.6 kJ, translating to 27600 J. This is the direct heat transfer during the process.
Key points to note:

• Use this formula to calculate how much heat is exchanged.
• Make sure to convert all units to match—joules for energy and grams for mass, for example.
• Understanding the context: knowing if a process is exothermic or endothermic can guide your calculation direction (i.e., increase or decrease in temperature).

These calculations help predict temperature changes resulting from chemical reactions or physical changes.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of a unit mass of a substance by one degree Celsius. It's an intrinsic property that varies among different substances.
Water, for instance, has a specific heat capacity of 4.18 J/g·°C, meaning it takes 4.18 Joules of energy to raise the temperature of 1 gram of water by 1°C. This high capacity plays a crucial role in everyday phenomena and also in scientific experiments.
In practical terms:

• A higher specific heat capacity means a substance can absorb more heat without drastically changing its temperature.
• Substances with lower specific heat can heat up or cool down rapidly.

By understanding this concept, one can predict how different substances will respond to thermal energy inputs in various processes.

Temperature Conversion

Temperature conversions are useful when comparing measurements taken in different units. The most common conversions involve Fahrenheit and Celsius, two widely used temperature scales.
For converting Celsius to Fahrenheit, use the formula:\[ T_{(°F)} = T_{(°C)} \times \frac{9}{5} + 32 \]This formula converts a temperature from degrees Celsius to degrees Fahrenheit. In the exercise, the initial and final temperatures needed conversion:

• The initial 30°C converts to 54°F.
• The final 56.4°C converts to approximately 133.5°F.

Key points for conversion:

• Always ensure you have the correct conversion formula for the scales you're dealing with.
• Check your calculations—small errors can lead to significant inaccuracies in scientific contexts.
• Understanding this allows seamless communication across different scientific communities that may prefer one scale over another.

Temperature conversion helps provide easily understandable data across various metric systems.