A family comes home from a long vacation with laundry to do...

A family comes home from a long vacation with laundry to do and showers to take. The water heater has been wrned off during the vacation. If the heater has a capacity of $50.0$ gallons and a $4800-\mathrm{W}$ heating clement, how much lime is required to raise the temperature of the water from $20.0^{-} \mathrm{C}$ to $60.0^{\circ} \mathrm{C}$ ? Assume the heater is well insulated and no water is withelrawn from the tank during that time.

Key Concepts

Unit Conversions

Often a heating problem involves converting from one set of units to another, such as gallons to liters or kilograms. Accurate unit conversions are essential to ensure that all quantities (volume, mass, energy) are compatible in the calculations.

System Insulation Assumptions

Assuming that a heater is well insulated implies that nearly all the energy delivered is used to heat the water rather than being lost to the environment. This simplifies the calculation by allowing focus solely on the energy transfer to the water.

Specific Heat Capacity

This concept refers to the amount of heat energy required to raise the temperature of a unit mass of a substance by one degree Celsius. In water heating problems, it is used to determine the total energy needed to achieve a desired change in temperature.

Energy Transfer Equation (Q = mc?T)

This equation connects energy, mass, specific heat capacity, and temperature change. It is fundamental for calculating the total energy required to increase the temperature of a substance, such as water in a heater.

Power and Energy Relationship

Power is the rate at which energy is transferred. Understanding that energy (in joules) divided by power (in watts) gives the time required to deliver that energy is crucial for determining how long a heating element must operate to achieve a desired temperature increase.

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