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 turned off during the vacation. If the heater has a capacity of 50.0 gallons and a $4800-\mathrm{W}$ heating element, how much time is required to raise the temperature of the water from $20.0^{\circ} \mathrm{C}$ to $60.0^{\circ} \mathrm{C}$ ? Assume the heater is well insulated and no water is withdrawn from the tank during that time.

Key Concepts

Unit Conversions

In many practical problems, converting between different units of measurement (such as volume to mass, or energy units) is essential. Accurate conversion factors ensure that calculations in thermal and energy problems remain consistent, especially when dealing with real-world quantities like water volume and mass.

Conservation of Energy

This principle states that energy in an isolated system is conserved, meaning that all energy supplied to a system (like a heater) goes into changing the internal energy, in this case, increasing the temperature of the water. It is a foundational concept in thermodynamics and ensures that energy calculations account for all energy input in a well-insulated system.

Energy Calculation

The energy required to change the temperature of a material can be calculated by multiplying its mass, its specific heat capacity, and the change in temperature. This fundamental relationship allows one to determine how much energy an appliance needs to supply to achieve a desired temperature change in a given substance.

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. It is a material-specific property that plays a central role in determining how much energy is needed for temperature changes, and is used in calculations involving the heating or cooling of substances.

Power

Power is the rate at which energy is transferred or converted, measured in watts (joules per second). In heating applications, it is used to determine how quickly a certain amount of energy can be delivered, thereby linking the energy requirement of a system to the time needed to achieve a temperature change.

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