Problem 200 - Introductory Chemistry Atoms First

Question

Calculate the amount of heat energy in joules required to heat \(50.0 \mathrm{~g}\) of each substance from \(25.0^{\circ} \mathrm{C}\) to \(37.0^{\circ} \mathrm{C}\). (specific heats shown in parentheses): (a) Iron \(\left(0.449 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\) (b) Aluminum (0.901 J/g \(\cdot{ }^{\circ} \mathrm{C}\) ) (c) Mercury \(\left(0.14 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\) (d) Water \(\left(4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ} \mathrm{C}\right)\)

Step-by-Step Solution

Answer

The heat energy required for each substance is: (a) Iron: \(269.4\mathrm{~J}\) (b) Aluminum: \(541.2\mathrm{~J}\) (c) Mercury: \(84.0\mathrm{~J}\) (d) Water: \(2510.4\mathrm{~J}\)

Step 1: Identify the given information for each substance

We have the following information for each substance: (a) Iron: mass \(m = 50.0\mathrm{~g}\), specific heat capacity \(c = 0.449 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ}\mathrm{C}\) (b) Aluminum: mass \(m = 50.0\mathrm{~g}\), specific heat capacity \(c = 0.901 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ}\mathrm{C}\) (c) Mercury: mass \(m= 50.0\mathrm{~g}\), specific heat capacity \(c = 0.14 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ}\mathrm{C}\) (d) Water: mass \(m= 50.0\mathrm{~g}\), specific heat capacity \(c = 4.18 \mathrm{~J} / \mathrm{g} \cdot{ }^{\circ}\mathrm{C}\)

Step 2: Calculate the temperature change for each substance

For each substance, we are given the initial temperature \(T_{i} = 25.0^{\circ}\mathrm{C}\) and the final temperature \(T_{f} = 37.0^{\circ}\mathrm{C}\). To calculate the temperature change, \(ΔT\), we use the formula \(ΔT = T_{f} - T_{i}\): \(ΔT = 37.0^{\circ}\mathrm{C} - 25.0^{\circ}\mathrm{C} = 12.0^{\circ}\mathrm{C}\)

Step 3: Calculate the heat energy required for each substance

Using the formula for heat energy \(q = mcΔT\), we can calculate the heat energy required for each substance: (a) Iron: \(q_{\text{iron}} = (50.0\mathrm{~g})(0.449 \frac{\mathrm{J}}{\mathrm{g} \cdot{ }^{\circ}\mathrm{C}})(12.0^{\circ}\mathrm{C}) = 269.4\mathrm{~J}\) (b) Aluminum: \(q_{\text{aluminum}} = (50.0\mathrm{~g})(0.901 \frac{\mathrm{J}}{\mathrm{g} \cdot{ }^{\circ}\mathrm{C}})(12.0^{\circ}\mathrm{C}) = 541.2\mathrm{~J}\) (c) Mercury: \(q_{\text{mercury}} = (50.0\mathrm{~g})(0.14 \frac{\mathrm{J}}{\mathrm{g} \cdot{ }^{\circ}\mathrm{C}})(12.0^{\circ}\mathrm{C}) = 84.0\mathrm{~J}\) (d) Water: \(q_{\text{water}} = (50.0\mathrm{~g})(4.18\frac{\mathrm{J}}{\mathrm{g} \cdot{ }^{\circ}\mathrm{C}})(12.0^{\circ}\mathrm{C}) = 2510.4\mathrm{~J}\) The heat energy required for each substance is: (a) Iron: \(269.4\mathrm{~J}\) (b) Aluminum: \(541.2\mathrm{~J}\) (c) Mercury: \(84.0\mathrm{~J}\) (d) Water: \(2510.4\mathrm{~J}\)

Key Concepts

Heat Energy

Heat energy, also known as thermal energy, is a form of energy transfer between substances or systems due to a temperature difference. In the context of the given exercise, heat energy is the amount of energy required to raise the temperature of a substance.

When a substance absorbs heat energy, it does not necessarily change its temperature instantly. Instead, the energy goes into increasing the kinetic energy of its particles, which may result in a temperature rise. It's important to note that different substances will require different amounts of heat energy to change their temperature by the same amount. This is largely due to differences in their specific heat capacities.

Temperature Change

Temperature change indicates the difference between the final temperature and the initial temperature of an object or substance. It is a valuable measure because it tells us how much a substance's temperature has increased or decreased.

The formula to calculate temperature change, represented as \( \Delta T \), is relatively straightforward: \( \Delta T = T_f - T_i \), where \( T_f \) is the final temperature and \( T_i \) is the initial temperature. In the exercise, the temperature of various substances is raised from 25.0°C to 37.0°C, resulting in a temperature change of 12.0°C for each substance.

Specific Heat Formula

The specific heat formula is fundamental in calculating the amount of heat energy required for a temperature change. The specific heat capacity (c) is a property that describes how much heat energy is needed to raise the temperature of one gram of a substance by one degree Celsius.

The formula to compute the heat energy absorbed or released is \( q = mc\Delta T \), wherein \( q \) represents the heat energy in joules, \( m \) is the mass of the substance in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.

Applying this to the exercise, different substances with their respective specific heat capacities used the same formula to determine the required heat energy for the given temperature change. For example, the heat energy for iron was calculated using the values \( m = 50.0\,\mathrm{g} \) and \( c = 0.449\,\mathrm{J/g\cdot{ }^{\circ}C} \), resulting in the calculation \( q_{\text{iron}} = (50.0\,\mathrm{g}) (0.449\,\mathrm{J/g\cdot{ }^{\circ}C}) (12.0^{\circ}\mathrm{C}) = 269.4\,\mathrm{J} \). Each substance's specific heat capacity influences how much heat energy is needed for the same temperature change, highlighting the importance of understanding this concept in thermal physics.