Question

As a \(0.1\) mole sample of solid \(\mathrm{NH}_{4} \mathrm{Cl}\) was dissolved in \(50 \mathrm{ml}\) of water, the temperature of the solution decreased. A small electrical immersion heater restored the temperature of the system by passing \(0.125 \mathrm{~A}\) from a \(15 \mathrm{~V}\) power supply for a period of 14 min. \(\Delta H\) for the process: \(\mathrm{NH}_{4} \mathrm{Cl}(\mathrm{s}) \rightarrow \mathrm{NH}_{4} \mathrm{Cl}(\mathrm{aq})\) is (a) \(-15.75 \mathrm{~kJ}\) (b) \(+15.75 \mathrm{~kJ}\) (c) \(-787.5 \mathrm{~J}\) (d) \(+787.5 \mathrm{~J}\)

Short answer

There is a mismatch in the values provided by the options and the calculated energy; the calculated energy supplied by the heater is \( +1575 \text{ J} \) or \( +1.575 \text{ kJ} \), but this doesn't match with any of the given options exactly.

Step-by-step solution

Calculate the energy used by the heater

To determine the energy supplied by the heater, use the formula for electrical power: \( P = I \times V \), where \( P \) is power, \( I \) is current, and \( V \) is voltage. The energy \( E \) can then be found by multiplying the power by the time in seconds: \( E = P \times t \).

Convert the time from minutes to seconds

Since power is often in terms of seconds, convert the heating time from minutes to seconds. There are 60 seconds in a minute, so multiply the time by 60: \( t = 14 \text{ min} \times 60 \text{ sec/min} = 840 \text{ sec} \text{.} \)

Calculate the power of the heater

Using the power formula, calculate the power of the heater: \( P = I \times V = 0.125 \text{ A} \times 15 \text{ V} = 1.875 \text{ W} \text{.} \)

Calculate the energy supplied by the heater

Now, calculate the energy supplied by the heater using the power and the time in seconds: \( E = P \times t = 1.875 \text{ W} \times 840 \text{ sec} = 1575 \text{ J} \text{.} \)

Determine the sign of \( \Delta H \) for the process

Given that the temperature of the solution decreased, the process is endothermic. Therefore, the system absorbed heat, and \( \Delta H \) for the process should be positive.

Compare with the provided options

Comparing the calculated energy with the options, the closest match is option (b) \( +1575 \text{ J} \) which we will have to convert to kJ by dividing by 1000. After converting, \( 1575 \text{ J} \) becomes \( +1.575 \text{ kJ} \). However, we must note that we have a discrepancy as the closest option given is (b) \( +15.75 \text{ kJ} \) suggesting a possible mistake in the options provided or in our calculation.

Key concepts

Thermochemistry

Thermochemistry is a branch of thermodynamics focused on the heat changes that occur during chemical and physical processes. It is essential for understanding the energy aspects of reactions taking place at constant pressure, which is the condition of most laboratory and everyday occurrences. In a thermochemical equation, the enthalpy change (\(\Delta H\))) accompanies the chemical reaction, and it is an important factor in determining whether a reaction is feasible under the given conditions.

When discussing the exercise involving the dissolution of \(\mathrm{NH}_{4} \mathrm{Cl}\) in water, we refer to the resultant \(\Delta H\) of the process. The enthalpy change can be measured experimentally by determining the amount of heat absorbed or released by the system. This is usually done by calculating the amount of energy required to reverse any temperature change in the reaction medium, which in this case was achieved using an electrical heater.

Endothermic Process

An endothermic process is one that absorbs heat energy from its surroundings. This causes the surroundings to cool down, as heat is essentially transferred from the environment to the system. According to the first law of thermodynamics, the change in internal energy (\(\Delta U\))) of a system is equal to the heat added to the system (\(Q\))) minus the work done by the system (\(W\))): \(\Delta U = Q - W\). For endothermic processes, the \(Q\) value is positive because heat is being absorbed.

In the example provided, the dissolution of \(\mathrm{NH}_{4} \mathrm{Cl}\) in water is an endothermic process because the temperature decreases, indicating that the solution absorbs heat as the salt dissolves. This absorption of heat is precisely what determines that \(\Delta H\) for the process is positive, as correctly identified in Step 5 of the step-by-step solution.

Electrical Energy Calculation

Calculating electrical energy is a crucial aspect of understanding how much work or heat can be provided by an electrical source. The fundamental relationship for electrical power (\(P\))) is given by the product of current (\(I\))) in amperes and voltage (\(V\))) in volts: \(P = I \times V\). To calculate the energy (\(E\))) consumed or delivered by an electrical device, one must multiply the power by the duration (\(t\))) for which the current flows, measured in seconds: \(E = P \times t\).

Using this relationship, the exercise demonstrates how to calculate the energy used by an electrical heater to restore the temperature of the solution affected by an endothermic process. It is this calculation that helps derive the enthalpy change of the dissolution process by converting the energy spent by the heater into joules or kilojoules, aligning it with the standard units used for \(\Delta H\).

Heats of Solution

The heat of solution, or enthalpy of solution, is the enthalpy change associated with the dissolution of a substance in a solvent. This value can be either positive or negative, representing an endothermic or exothermic process, respectively. The heat of solution is a crucial concept in understanding how substances interact with solvents and involves measuring the heat exchange involved when a solute dissolves.

During the problem-solving process, one must consider that the amount of solute and the volume of solvent can affect the magnitude of \(\Delta H\). In the given exercise, the dissolution of a specific amount of \(\mathrm{NH}_{4} \mathrm{Cl}\) leads to a quantifiable change in temperature that was countered by the heat supplied by an electric heater. Thus, the calculation of the heat of solution in this instance directly relates to the heat necessary to maintain the temperature of the aqueous solution, underscoring the connection between the electrical energy supplied and the resulting enthalpy change.