Problem 55 - Chemistry The Science in Context

Question

Exactly \(10.0 \mathrm{mL}\) of water at \(25.0^{\circ} \mathrm{C}\) is added to a hot iron skillet. All the water is converted into steam at \(100.0^{\circ} \mathrm{C}\) The mass of the pan is \(1.20 \mathrm{kg}\) and the molar heat capacity of iron is \(25.19 \mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right) .\) What is the temperature change of the skillet?

Step-by-Step Solution

Answer

Answer: The temperature change of the skillet is \(104.3^{\circ} \mathrm{C}\).

Step 1: Identify given values and constants

The following values and constants are given in the problem: Initial water temperature: \(T_{wi} = 25^{\circ} \mathrm{C}\) Final water temperature: \(T_{wf} = 100^{\circ} \mathrm{C}\) Water volume: \(V_w = 10.0 \mathrm{mL}\) Mass of skillet: \(m_{s} = 1.20 \mathrm{kg}\) Molar heat capacity of iron: \(C_{m,Iron} = 25.19 \mathrm{J} /\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\)

Step 2: Calculate the mass of water, the number of moles, and molar heat capacities of the substances

Assuming the density of water to be \(1 \mathrm{g}/\mathrm{mL}\), the mass of water can be found: \(m_w = V_w \times \rho_w = 10.0 \mathrm{mL} \times 1 \mathrm{g}/\mathrm{mL} = 10.0 \mathrm{g}\) Convert the mass to kg: \(m_w = 10.0 \mathrm{g} \times \frac{1 \mathrm{kg}}{1000 \mathrm{g}} = 0.01 \mathrm{kg}\) The molar heat capacity for water is given by \(C_p,H2O = 75.3 \mathrm{J}/\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)\). We also need to calculate the molar heat capacity of the skillet in \(\mathrm{J}/\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\). Using the molar mass of iron, \(M_{Iron} = 55.845\mathrm{g}/\mathrm{mol}\), and converting it to kg/mol, the molar heat capacity for the skillet can be calculated: \(C_{p,Iron} = \frac{C_{m,Iron}}{M_{Iron}} = \frac{25.19 \mathrm{J}/\left(\mathrm{mol} \cdot^{\circ} \mathrm{C}\right)}{55.845\mathrm{g}/\mathrm{mol} \times \frac{1 \mathrm{kg}}{1000 \mathrm{g}}} = 0.4509 \mathrm{J}/\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)\)

Step 3: Calculate the heat absorbed by water

To calculate the heat absorbed by water (Q), we'll use the following equation: \(Q_w = m_w \times C_p,H2O \times \Delta T_w\) \(\Delta T_w = T_{wf} - T_{wi} = 100^{\circ} \mathrm{C} - 25^{\circ} \mathrm{C} = 75^{\circ} \mathrm{C}\) \(Q_w = 0.01 \mathrm{kg} \times 75.3 \mathrm{J}/\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right) \times 75^{\circ} \mathrm{C} = 56.475 \mathrm{J}\)

Step 4: Calculate the temperature change of the skillet

As the heat absorbed by the water is equal to the heat lost by the skillet, we can calculate the temperature change of the skillet using the formula: \(\Delta T_s = \frac{Q_w}{m_s \times C_{p,Iron}}\) \(\Delta T_s = \frac{56.475 \mathrm{J}}{1.20 \mathrm{kg} \times 0.4509 \mathrm{J}/\left(\mathrm{kg} \cdot^{\circ} \mathrm{C}\right)} = 104.3^{\circ} \mathrm{C}\) The temperature change of the skillet is \(104.3^{\circ} \mathrm{C}\).

Key Concepts

Heat Capacity

Heat capacity is a measure of the amount of heat energy required to raise the temperature of a body by a certain degree. It is a bulk property of a material and depends on the amount of substance present. When we talk about heat capacity, it's important to differentiate it from specific heat. Heat capacity is the total heat needed to change the temperature of an entire object, regardless of its size or mass, whereas specific heat is the heat capacity per unit mass.

For instance, a larger iron skillet will have a higher heat capacity than a smaller one simply because there is more iron to heat up. This means that more heat energy would be required to raise the temperature of the larger skillet by the same amount as the smaller one. In thermal calculations, heat capacity plays a crucial role as it helps in determining how much energy transfer will result in a particular temperature change for a given substance.

Specific Heat Calculation

The specific heat of a substance is a measure of how much energy, in the form of heat, is needed to raise the temperature of one kilogram of the substance by one degree Celsius (\textdegree C). It is denoted by the symbol 'Cp' and commonly expressed in units of Joules per kilogram per degree Celsius (\text{J/(kg·\textdegree C)}).

To calculate the specific heat in a problem, you can use the formula: \[C_p = \frac{Q}{m \times \Delta T}\]where:

• \(Q\) is the total heat added or removed (in Joules),
• \(m\) is the mass of the substance (in kilograms),
• \(\Delta T\) is the change in temperature (\textdegree C).

By rearranging the formula, you can also calculate the heat transferred if you know the specific heat, mass, and temperature change: \[Q = m \times C_p \times \Delta T\]This is fundamental in thermochemistry problems, and understanding this concept allows students to compute the energy involved in heating or cooling a substance.

Energy Transfer

Energy transfer in the form of heat occurs when there's a temperature difference between two bodies or systems; energy will flow from the hotter to the cooler object until thermal equilibrium is reached. This process can be quantified using the concept of heat (Q), which in the context of thermochemistry is often expressed in Joules (J).

The amount of energy transferred depends on the material's properties (specific heat capacity) and the mass and temperature change of the substance involved. By knowing these factors, we can use the relationship \[Q = m \times C_p \times \Delta T\]to calculate the exact amount of heat that is transferred in a reaction or between objects. It's crucial to note that energy is conserved in these interactions, so the heat lost by one body is equal to the heat gained by another, assuming no losses to the surroundings. This principle is widely used in calorimetry to determine specific heats and other thermal properties of substances.

Temperature Change

Temperature change is a direct indicator of energy transfer in a system and is expressed in degrees Celsius (\textdegree C) or Kelvin (K). When a substance absorbs or loses heat, its temperature will respectively rise or fall; this change can be calculated if the amount of heat and the specific heat capacity are known. The formula \[\Delta T = \frac{Q}{m \times C_p}\]allows us to find the change in temperature when a known amount of heat is added to or removed from a given mass of a substance with a specific heat capacity.

In the described exercise, we use this principle to determine the temperature change of an iron skillet when water is heated from 25\textdegree C to 100\textdegree C and turned into steam. It's a clear demonstration of how energy is transferred between different substances and how their temperatures are affected as a result.