Problem 30 - General Chemistry

Question

a. A \(100 .-\mathrm{g}\) sample of water is placed in an insulated container and allowed to come to room temperature at \(21^{\circ} \mathrm{C}\). To heat the water sample to \(41^{\circ} \mathrm{C}\), how much heat must you add to it? b. Consider the hypothetical reaction, $$ 2 \mathrm{X}(a q)+\mathrm{Y}(l) \longrightarrow \mathrm{X}_{2} \mathrm{Y}(a q) $$ being run in an insulated container that contains \(100 . \mathrm{g}\) of solution. If the temperature of the solution changes from \(21^{\circ} \mathrm{C}\) to \(31^{\circ} \mathrm{C}\), how much heat does the chemical reaction produce? How does this answer compare with that in part a? (You can assume that this solution is so dilute that it has the same heat capacity as pure water.) C. If you wanted the temperature of \(100 . \mathrm{g}\) of this solution to increase from \(21^{\circ} \mathrm{C}\) to \(51^{\circ} \mathrm{C}\), how much heat would you have to add to it? (Try to answer this question without using a formula.) d. If you had added \(0.02\) mol of \(X\) and \(0.01\) mol of \(Y\) to form the solution in part b, how many moles of \(\mathrm{X}\) and \(\mathrm{Y}\) would you need to bring about the temperature change described in part c. e. Judging on the basis of your answers so far, what is the enthalpy of the reaction \(2 \mathrm{X}(a q)+\mathrm{Y}(l) \longrightarrow \mathrm{X}_{2} \mathrm{Y}(a q) ?\)

Step-by-Step Solution

Answer

To heat the water to 41°C, add 8360 J. The reaction in part b produces 4180 J. Twice the moles (0.04 mol X, 0.02 mol Y) are needed for the 30°C rise, and the reaction's enthalpy is 418 kJ/mol.

Step 1: Calculate Heat Needed for Water Heating

To determine how much heat is required to raise the temperature of 100 g of water from 21°C to 41°C, we use the formula:\[q = m \cdot c \cdot \Delta T\]where - \( m \) is the mass (100 g), - \( c \) is the specific heat capacity of water (4.18 J/g°C), - \( \Delta T \) is the change in temperature (41°C - 21°C = 20°C).Substituting the values:\[q = 100 \times 4.18 \times 20 = 8360 \text{ J}\]

Step 2: Calculate Heat from Chemical Reaction

The reaction causes the temperature of the solution to rise from 21°C to 31°C. Again, we use the formula for heat (assuming the same heat capacity as water):\[q = m \cdot c \cdot \Delta T\]where - \( m \) is 100 g,- \( c \) is the heat capacity (4.18 J/g°C), - \( \Delta T \) is 31°C - 21°C = 10°C.Substituting the values:\[q = 100 \times 4.18 \times 10 = 4180 \text{ J}\]This heat is produced by the reaction.

Step 3: Compare Heat from Parts a and b

The heat required in part a is 8360 J, while the heat produced in part b is 4180 J. The heat produced in part b is half of what is needed to raise the temperature to 41°C.

Step 4: Heat Required for Further Increase to 51°C

Since the heat required to increase temperature from 21°C to 31°C was 4180 J and knowing it will take a similar amount for the next 10°C, the total heat needed to increase from 21°C to 51°C is:\[2 \times 4180 = 8360 \text{ J}\]

Step 5: Determine Moles of Reactants Needed

From part b, 0.02 mol of X and 0.01 mol of Y produce 4180 J. To produce 8360 J (as in part c), twice the moles are required: - 0.04 mol of X, - 0.02 mol of Y.

Step 6: Calculate Enthalpy Change for Reaction

The enthalpy change, \( \Delta H \), is the energy change per mole of reaction. If 4180 J corresponds to 0.01 mol of reaction, then:\[\Delta H = \frac{4180}{0.01} = 418000 \text{ J/mol} = 418 \text{ kJ/mol}\]

Key Concepts

Heat capacity

Heat capacity is an essential topic in thermodynamics, playing a key role in determining the amount of energy needed to change the temperature of a substance. Simply put, it is a measure of the heat energy required to increase the temperature of a certain amount of a substance by a given amount, usually 1 degree Celsius or 1 Kelvin. For any material, it depends heavily on its specific characteristics, such as the type of bonds, structure, and state (solid, liquid, gas).

Heat capacity is denoted by the symbol "C" and is measured in joules per degree Celsius (J/°C) or joules per Kelvin (J/K). The larger the heat capacity of a substance, the more energy is needed to produce a temperature change. In our example, the heat capacity of water is considered to calculate how much heat is needed to raise the sample's temperature. Understanding heat capacity is crucial when performing energy calculations in experiments and industrial processes.

Enthalpy change

Enthalpy change, represented as \( \Delta H \), is a central concept in thermodynamics and chemistry. It refers to the heat absorbed or released in a chemical reaction at constant pressure. Importantly, enthalpy reflects the total energy change in a system, encompassing both internal energy shifts and work done by the system.

In a chemical reaction, calculating the enthalpy change can reveal whether a process is endothermic (requiring heat input) or exothermic (releasing heat). For instance, when analyzing a hypothetical reaction in our exercise, we aim to determine how much heat the reaction produces and how it compares to other scenarios. This involves measuring the heat absorbed or evolved as reactants convert into products.

Knowing the enthalpy change is vital for predicting reaction behavior and designing processes to control energy changes effectively.

Specific heat capacity

Specific heat capacity is a specialized form of heat capacity that describes how much energy is required to raise the temperature of one gram of a substance by one degree Celsius. It is represented by the symbol "c" and is expressed in terms of J/g°C or J/gK.

Water's specific heat capacity, for instance, is 4.18 J/g°C, indicating that each gram of water requires 4.18 joules to increase its temperature by one degree Celsius. This property is critical in determining energy changes when heating or cooling a substance and helps in calculating the thermal energy changes in various chemical processes.

In our exercise, the specific heat capacity of water is used to find out how much heat energy is needed to raise the temperature of our water sample from 21°C to 41°C. Specific heat capacity provides a straightforward way to calculate energy requirements for temperature changes in substances.

Chemical reactions

Chemical reactions are transformations where reactants convert into products, involving changes in energy. During these reactions, bonds are broken and formed, leading to different arrangements of atoms.

When discussing the energy aspects of chemical reactions, it's essential to consider the heat involved, which can be measured or calculated as an enthalpy change. Some reactions absorb heat (endothermic), while others release heat (exothermic).

In our example, a hypothetical reaction in an insulated container causes a temperature rise from 21°C to 31°C. This tells us the reaction is exothermic as it releases heat into the surroundings, raising the temperature of the solution. Understanding chemical reactions involves not just balancing equations but also comprehending the energetic changes they cause.

Energy calculations

Energy calculations in thermodynamics involve determining the amount of heat exchanged in physical processes and chemical reactions. These calculations help us quantify how much energy is absorbed or released, which is vital for applications ranging from engine design to everyday appliances.

When it comes to heating, cooling, or chemical transformations, the formula \( q = m \cdot c \cdot \Delta T \) allows us to calculate heat exchange. Here, \( q \) is the heat exchanged in joules, \( m \) is the mass in grams, \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.
In educational settings, understanding energy calculations equips students with the ability to analyze experiments and predict the energy requirements of chemical processes. This foundational knowledge is crucial for scientists and engineers who work on energy management in various fields.