Question

Glauber's salt, sodium sulfate decahydrate \(\left(\mathrm{Na}_{2} \mathrm{SO}_{4} .\right.\) \(\left.10 \mathrm{H}_{2} \mathrm{O}\right),\) undergoes a phase transition (i.e., melting or freezing) at a convenient temperature of about \(32^{\circ} \mathrm{C}\) : \(\begin{aligned}{\mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}(s) \longrightarrow \mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O}(l)}{\Delta H^{\circ}} &=74.4 \mathrm{~kJ} / \mathrm{mol} \end{aligned}\) As a result, this compound is used to regulate the temperature in homes. It is placed in plastic bags in the ceiling of a room. During the day, the endothermic melting process absorbs heat from the surroundings, cooling the room. At night, it gives off heat as it freezes. Calculate the mass of Glauber's salt in kilograms needed to lower the temperature of air in a room by \(8.2^{\circ} \mathrm{C}\). The mass of air in the room is \(605.4 \mathrm{~kg} ;\) the specific heat of air is \(1.2 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\).

Short answer

You need 25.87 kg of Glauber's salt.

Step-by-step solution

Identify the Problem Components

The problem involves using the latent heat from the phase transition of Glauber's salt to lower the temperature of air. We are given: 1. The phase transition enthalpy \( \Delta H^{\circ} = 74.4 \text{ kJ/mol} \) for Glauber's salt.2. The mass of the air in the room (605.4 kg).3. The specific heat of air (1.2 J/g°C).4. The temperature change required (8.2°C).

Calculate the Heat needed to Cool the Air

First, convert the mass of air to grams: \( 605.4 \text{ kg} = 605400 \text{ g} \).Then, calculate the heat needed using the formula:\[ q = m \cdot c \cdot \Delta T \]where \( m = 605400 \text{ g} \), \( c = 1.2 \text{ J/g°C} \), and \( \Delta T = 8.2^{\circ} \text{C} \).Substitute the values:\[ q = 605400 \text{ g} \times 1.2 \text{ J/g°C} \times 8.2^{\circ} \text{C} \]

Perform the Heat Calculation

Calculate the heat energy required to lower the temperature:\[ q = 605400 \text{ g} \times 1.2 \times 8.2 \]\[ q = 5978688 \text{ J} = 5978.688 \text{ kJ} \].

Determine Moles of Glauber's Salt Needed

Given that the enthalpy change for the phase transition is \( 74.4 \text{ kJ/mol} \), calculate the number of moles required to absorb \( 5978.688 \text{ kJ} \):\[ n = \frac{5978.688 \text{ kJ}}{74.4 \text{ kJ/mol}} \].

Calculate the Mass of Glauber's Salt

Calculate the number of moles:\[ n = \frac{5978.688}{74.4} \approx 80.3 \text{ mol} \].The molar mass of Glauber's salt \( \mathrm{Na}_{2} \mathrm{SO}_{4} \cdot 10 \mathrm{H}_{2} \mathrm{O} \) is 322.2 g/mol.Calculate the mass:\[ \text{mass} = 80.3 \times 322.2 \text{ g/mol} \approx 25867.86 \text{ g} \].

Convert Mass to Kilograms

Convert the mass from grams to kilograms to find the mass of Glauber's salt needed:\[ 25867.86 \text{ g} = 25.87 \text{ kg} \].

Key concepts

Phase Transition

Imagine you have an ice cube melting into water. This is an example of a phase transition. It's the process where a substance changes from one state of matter (like solid) to another (like liquid). In the case of Glauber's salt, it transitions from a solid to a liquid at around 32°C. These changes usually involve heating (melting) or cooling (freezing) as heat energy is absorbed or released. This is crucial for temperature regulation, like in our exercise where Glauber's salt absorbs heat from the room, helping to cool it down during the day. At night, it solidifies and releases that heat back, warming the room.

Latent Heat

Latent heat is the amount of energy absorbed or released by a substance during its phase change without changing its temperature. Let's consider the melting of Glauber's salt. The energy needed for this transformation is known as latent heat. It is not used to increase the temperature but to facilitate the phase change itself. For Glauber's salt, we know that 74.4 kJ/mol is required for melting. This energy allows the salt to absorb heat when it melts, which is then released back when it solidifies, creating a natural temperature regulation in a room.

Enthalpy Change

When Glauber's salt undergoes a phase transition, there is a change in enthalpy (\(\Delta H^{\circ}\)). Enthalpy is a measurement of energy in a system, pertinent to heat. The enthalpy change is the difference in energy between the initial and final states of a phase transition. For Glauber's salt, the enthalpy change of 74.4 kJ/mol indicates the energy exchange during melting. This value is crucial for calculating how much heat the salt can absorb from or release to the environment, playing a key role in its ability to regulate room temperature.

Specific Heat Capacity

Specific heat capacity is the amount of heat required to change the temperature of one gram of a substance by one degree Celsius. In the example of the air in the room, the specific heat capacity is given as 1.2 J/g°C. This tells us how much heat the air can hold. When we want to change the air temperature by a certain amount, knowing its specific heat helps us calculate how much total heat is required. In our exercise, this specific heat helps determine the total energy that Glauber's salt needs to absorb to lower the room’s temperature by 8.2°C.