Question

Water is boiled at 1 atm pressure in a coffee maker equipped with an immersion-type electric heating element. The coffee maker initially contains 1 kg of water. Once boiling started, it is observed that half of the water in the coffee maker evaporated in 10 minutes. If the heat loss from the coffee maker is negligible, the power rating of the heating element is \((a) 3.8 \mathrm{kW}\) (b) \(2.2 \mathrm{kW}\) \((c) 1.9 \mathrm{kW}\) \((d) 1.6 \mathrm{kW}\) \((e) 0.8 \mathrm{kW}\)

Short answer

Answer: The power rating of the electric heating element is approximately 1.9 kW.

Step-by-step solution

Identify the variables

Here are the given variables: - mass of water initially (m1) = 1 kg - mass of water evaporated (m2) = 0.5 kg - time taken for evaporation (t) = 10 minutes = 600 seconds - Pressure = 1 atm - Specific heat of water (c) = 4.186 J/g°C [standard] - Latent heat of vaporization for water, Lv = 2256 J/g [1 atm]

Calculate the energy required for vaporization

We need to find the energy (Q) required to vaporize 0.5 kg of water at 1 atm pressure. To get this, we will use the formula: \(Q = m_{2} \cdot L_{v}\) Where \(m_{2}\) is the mass of water evaporated, and \(L_{v}\) is the latent heat of vaporization. Remember to convert the mass from kg to grams (g) before calculating.

Convert mass to grams

Since the mass of water evaporated is given in kg and the latent heat is given in J/g, we have to convert the mass to grams: 1 kg = 1000 g 0.5 kg = 500 g

Calculate the energy (Q)

Substitute the values into the formula and calculate the energy required: \(Q = (500 \,\text{g}) \cdot (2256 \frac{\text{J}}{\text{g}}) = 1128000 \, \text{J}\)

Calculate the power rating of the heating element (P)

Now, to find the power (P) of the heating element, use the formula: \(P = \frac{Q}{t}\) Where \(t\) is the time taken for evaporation (in seconds). Substitute the values into the formula and get the power rating: \(P = \frac{1128000 \, \text{J}}{600\,s} = 1880 \,\text{W} = 1.88 \,\text{kW}\) The power rating of the heating element is \(\approx 1.9 \,\text{kW}\), which aligns with the answer (c).

Key concepts

Energy Transfer in Thermodynamics

Understanding energy transfer in thermodynamics is essential to grasp why certain processes, like boiling water in a coffee maker, consume a specific amount of power. In the context of our exercise, energy is transferred from the electric heating element to the water, increasing the water’s energy until it changes from liquid to vapor. This process is governed by the first law of thermodynamics, which states that energy can neither be created nor destroyed, only transformed.

Here, the transfer takes the form of heat, specifically the latent heat of vaporization, which is the amount of energy needed to convert water from its liquid phase to its gaseous phase without changing its temperature. The power rating of the heating element reflects how much energy per unit of time is supplied to achieve this phase change.

• Latent heat: Total energy needed for the phase change
• Energy transfer: Process of moving energy from the heating element to the water
• First law of thermodynamics: Principle of energy conservation in any process

Heat Capacity

Heat capacity is a property that dictates how much heat energy is required to raise the temperature of a given amount of a substance by a certain temperature interval. It’s an intrinsic characteristic that varies between different materials. In our context, the specific heat capacity of water at 4.186 J/g°C illustrates that water requires a significant amount of energy to change temperature.

This concept is separate from the problem at hand, as the focus is on the energy required for a phase change and not a temperature change. However, understanding heat capacity is crucial when considering why different substances heat up at different rates, a fundamental principle in thermodynamics.

Key Points to Remember:

• Heat capacity relates to temperature change, not phase change.
• Specific heat capacity (c) is the amount of heat per mass unit required to raise the temperature by one degree Celsius.
• Substances with high heat capacities require more energy to change temperature.

Thermodynamics of Phase Change

The thermodynamics of phase change is a critical concept that provides insight into how substances transition between different states of matter: solid, liquid, and gas. This phase change occurs at constant pressure and temperature and involves the transfer of energy without altering the temperature—a process called an isothermal process.

In the given exercise, the phase change from liquid to gas (evaporation) involves the latent heat of vaporization, Lv. As water boils in the coffee maker, it absorbs energy equivalent to its latent heat, allowing the molecules to break free from the liquid's surface and spread out as vapor.

Considerations During Phase Change:

• Phase changes are isothermal processes (occur at constant temperature).
• Latent heat is involved, which is the heat required for changing the phase at constant temperature.
• The amount of energy needed is determined by the mass undergoing the phase change and the substance's latent heat.

Each phase has characteristic energy and entropy values, and the phase change embodies the competition between these values as a substance absorbs or releases latent heat.