A coffee maker can be modeled as a heating element...

A coffee maker can be modeled as a heating element (resistance $R$ ) connected to the outlet voltage of $120 \mathrm{V}$ (assumed to be dc). The heating element boils small amounts of water at a time as it brews the coffee. When bubbles of water vapor form, they carry liquid water up through the tubing. Because of this, the coffee maker boils $5.0 \%$ of the water that passes through it; the rest is heated to $100^{\circ} \mathrm{C}$ but remains liquid. Starting with water at $10^{\circ} \mathrm{C},$ the coffee maker can brew $1.0 \mathrm{L}$ of coffee in 8.0 min. Find the resistance $R$.

Key Concepts

Latent Heat of Vaporization

This concept refers to the amount of energy required for a substance to change its phase from liquid to vapor at its boiling point, without changing its temperature. In thermal problems involving phase change, such as boiling water, it is essential to account for the latent heat to correctly determine the total energy required when a portion of the water is vaporized.

Specific Heat Capacity

Specific heat capacity is the amount of energy required to raise the temperature of one unit mass of a substance by one degree Celsius. This concept is crucial to determining the energy needed to heat water from an initial temperature to its boiling point, without accounting for phase change.

Energy Balance in Thermal Systems

The energy balance involves equating the energy input, usually in the form of electrical energy converted to heat, to the energy required to heat a substance and possibly perform a phase change. In designing systems like a coffee maker, this conservation of energy principle helps determine device parameters, such as the necessary resistance to achieve the desired heating performance.

Electrical Power and Joule Heating

This concept involves the conversion of electrical energy into thermal energy through resistance. When a voltage is applied across a resistor, the resistor converts electrical energy into heat, following the relation P = V²/R. This process, known as Joule heating, is fundamental in understanding how electrical devices like heating elements work.

Resistance in Electric Circuits

Resistance is the property of a material that resists the flow of electric current, and it plays a key role in defining how much power is dissipated as heat. The relation V = IR and the power equation P = V²/R are critical in determining the resistor’s behavior under a specified voltage, which is essential when designing or analyzing electrical heating elements.

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