Question

A British thermal unit (Btu) is defined as the quantity of heat required to change the temperature of 1 lb of water by \(1^{\circ}\) F. Assume the specific heat of water to be independent of temperature. How much heat is required to raise the temperature of the water in a 40 gal water heater from 48 to \(145^{\circ} \mathrm{F}\) in \((\mathrm{a}) \mathrm{Btu}\) (b) kcal; (c) kJ?

Short answer

The required heat to raise the temperature of the water in a 40 gal water heater from 48 to \(145^{\circ} \mathrm{F}\) is \(32379.2 \, \mathrm{Btu}\), \(8163.5 \, \mathrm{kcal}\), or \(34139.8 \, \mathrm{kJ}\)

Step-by-step solution

Convert Volume to Weight

To find the weight of the water, multiply the volume by the density of the water. So, \(40 \, \mathrm{gal} \times 8.34 \, \mathrm{lb/gal} = 333.6 \, \mathrm{lb}\)

Find Temperature Difference

Subtract the initial temperature from the final temperature to find the difference. So, \(145^{\circ} \mathrm{F} - 48^{\circ} \mathrm{F} = 97^{\circ} \mathrm{F}\)

Calculate Heat in Btu

Insert the values into the heat formula \(Q = m \cdot c \cdot \Delta T\) where \(m = 333.6 \, \mathrm{lb}\), \(c = 1 \, \mathrm{Btu/lb^{\circ}F}\), and \(\Delta T = 97^{\circ} \mathrm{F}\). Therefore, \(Q = 333.6 \, \mathrm{lb} \times 1 \, \mathrm{Btu/lb^{\circ} F} \times 97^{\circ} \mathrm{F} = 32379.2 \, \mathrm{Btu}\).

Convert Btu to kcal and kJ

Use the conversion factors to convert Btu to kcal and kJ. So, for kcal: \(32379.2 \, \mathrm{Btu} \times 0.252 \, \mathrm{kcal/Btu} = 8163.5 \, \mathrm{kcal}\) and for kJ: \(32379.2 \, \mathrm{Btu} \times 1.055 \, \mathrm{kJ/Btu} = 34139.8 \, \mathrm{kJ}\)

Key concepts

British thermal unit

A British thermal unit, abbreviated as Btu, is a unit of heat energy. It represents the amount of heat required to raise the temperature of one pound of water by one degree Fahrenheit. This unit is commonly used in the United States to measure heating or cooling energy for systems like heaters and air conditioners.
Btu is a practical measure because it directly relates to the heat needed for typical tasks like heating water. In our exercise, to find out how much heat is needed to change the temperature of water, we utilize the specific heat of water (which is 1 Btu/lb°F). This means that for each pound of water, 1 Btu is needed to raise the temperature by 1°F.

When calculating heat, remember:

• Volume to Weight Conversion: First, convert the water volume into weight. As water is approximately 8.34 pounds per gallon, you multiply your total gallons to get the weight in pounds.
• Temperature Difference: Determine how many degrees the water's temperature needs to be increased.
• Heat Calculation: Multiply the weight of water by the specific heat and the temperature difference.

By following these steps, you can calculate the required heat in Btus easily.

Specific heat

Specific heat is a crucial concept in heat calculation. It defines the amount of heat per unit mass required to raise the temperature by a single degree. In simple terms, it tells you how much energy you need to change the temperature of a material.

Water is often used as a standard for specific heat because of its abundant use and reliable properties. The specific heat of water is 1 Btu/lb°F. This specific heat is assumed constant for practical calculations over small temperature ranges, even though it can vary slightly with temperature changes.

• Importance: Helps in determining how much energy is needed for temperature changes.
• Application: Essential in calculations involving heating and cooling systems.
• Units: Typically expressed in units like Btu/lb°F, Joules/kg°C, or calories/g°C.
  

Specific heat simplifies calculations by providing a straightforward way to determine energy transfer for heating or cooling processes.

Temperature conversion

Temperature conversion is essential when dealing with heat calculations in different systems of measurement. Temperatures can be represented in Fahrenheit, Celsius, or Kelvin. When calculating energy requirements, it's crucial to ensure that the temperatures used in calculations are in consistent units.
While our original problem deals with Fahrenheit, understanding how to convert between different units can be invaluable:

• To convert from Fahrenheit to Celsius: Subtract 32 from the Fahrenheit value, then multiply by 5/9.
• To convert Celsius to Fahrenheit: Multiply the Celsius value by 9/5, then add 32.

Even though the original calculation is in Fahrenheit, knowing how to switch between units can help you apply similar methods across various scenarios.

Energy conversion

Energy conversion deals with transforming energy from one unit to another. When working with heat energy, it's common to convert Btu to other energy units such as kilocalories or kilojoules. Conversions are often necessary because different regions or applications might prefer or require different units. For example:

• Converting Btu to kilocalories: Multiply the energy in Btus by 0.252 to get the energy in kilocalories.
• Converting Btu to kilojoules: Multiply the energy in Btus by 1.055 to obtain energy in kilojoules.
  

These conversions let you work with global standards or specific preferences. Understanding these conversions is particularly useful in situations where multiple measurement systems are in use or when communicating findings internationally.