A British thermal unit (Btu) is defined as the quantity of...

A British thermal unit (Btu) is defined as the quantity of heat required to change the temperature of 1 lb of water by $1^{\circ} \mathrm{F} .$ Assume the specific heat capacity 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 (a) $\mathrm{Btu}$; (b) $\mathrm{kcal} ;$ (c) kJ?

Key Concepts

Mass and Volume Conversion

In many practical applications, the substance’s mass must be determined from its volume, especially when working with liquids like water. For water, a commonly used conversion factor is that one gallon is approximately equivalent to 8.34 pounds, enabling the use of specific heat capacity in mass-based energy calculations.

Temperature Change

The temperature change is the difference between the final and initial temperatures. This change, when multiplied by the mass and the specific heat capacity, gives the total heat required. It is a critical step in any thermal energy calculation.

Energy Unit Conversion

Different energy units such as British Thermal Units (Btu), kilocalories (kcal), and kilojoules (kJ) are used to measure heat energy. Converting between these units using standard conversion factors is essential for expressing the result in the desired unit of measure.

Specific Heat Capacity

This concept refers to the amount of heat required to raise the temperature of one unit of mass of a substance by one degree. It is fundamental to calculations in thermodynamics that involve determining how much energy is needed to change the temperature of a given substance.

Transcript

00:01 In this question, we are looking for how much heat is required to raise a certain volume of water by a certain delta t.

00:08 So our volume of water is equal to 40 gallons.

00:16 And we know that 4 .55 liters are in one gallon.

00:24 So our volume in liters is 182 liters.

00:28 I'm just going to change this into milliliters.

00:31 We have 182 times 10 to the third milliliters of our water.

00:40 They tell us that our t initial is equal to 48 degrees fahrenheit.

00:46 We want to turn this into degrees celsius.

00:50 So we're going to take 5 divided by 9, take that temperature 48 minus 32 degrees celsius, which will equal 8 .89 degrees celsius.

01:03 They tell us that the t final is equal to 145 degrees fahrenheit.

01:09 So we're just going to do the same.

01:11 Turn it into degrees celsius, 145 minus 32 degrees celsius, which will equal 62 .78 degrees c.

01:22 We know that our density of water is equal to 1 gram per milliliter, and that the specific heat of our water is equal to 4 .186 joules per gram degree celsius.

01:39 So they are wanting us to find q in different units, so they want it in british thermal units, k -cal and kilojoules.

01:50 So before we can find our heat, we need to find our mass of our h -2o.

01:55 So our mass of h2o is equal to our volume, the density.

02:01 So 182 times 10 to the third milliliters times one gram per milliliter.

02:13 So our mass of our water is 182 times 10 to the third grams.

02:21 So we need to find delta t.

02:24 So delta t is our t final minus t initial...

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