You want to heat the air in your house with natural gas (CH4) . Assume your house has 275 m^2 (a

You want to heat the air in your house with natural gas...

You want to heat the air in your house with natural gas $\left(\mathrm{CH}_{4}\right) .$ Assume your house has $275 \mathrm{m}^{2}$ (about $2800 \mathrm{ft}^{2}$ ) of floor area and that the ceilings are $2.50 \mathrm{m}$ from the floors. The air in the house has a molar heat capacity of $29.1 \mathrm{J} / \mathrm{mol} \cdot \mathrm{K}$. (The number of moles of air in the house can be found by assuming that the average molar mass of air is $28.9 \mathrm{g} / \mathrm{mol}$ and that the density of air at these temperatures is $1.22 \mathrm{g} / \mathrm{L} .$ ) What mass of methane do you have to burn to heat the air from $15.0^{\circ} \mathrm{C}$ to $22.0^{\circ} \mathrm{C} ?$

Key Concepts

Density and Volume Conversions

Density and volume conversions are critical when determining the mass or the number of moles of a substance present in a given volume. By understanding the relationship between density (mass per unit volume) and molar mass, one can convert measurements of volume into moles, which is necessary for applying thermodynamic equations involving molar properties.

Fuel Energy Density and Combustion Stoichiometry

Fuel energy density and combustion stoichiometry relate to the energy released when a fuel like methane is burned. Energy density tells us the amount of energy produced per unit mass of fuel, and understanding stoichiometry helps in determining the necessary fuel mass to achieve a desired energy output. This concept is essential in applications where fuel is used to generate heat, ensuring that the energy input meets the energy demand.

Molar Heat Capacity

Molar heat capacity is a property that indicates how much heat energy is required to raise the temperature of one mole of a substance by one degree Kelvin. It is a fundamental concept in thermodynamics, allowing us to calculate the energy required to change the temperature of gases or liquids when the number of moles is known.

Thermal Energy Calculation

Thermal energy calculation involves determining the amount of energy needed to raise the temperature of a substance by a specified amount. This is typically done by multiplying the number of moles, the molar heat capacity, and the change in temperature. This concept is crucial in understanding how energy inputs can be translated into temperature changes in a system.

Transcript

00:01 Okay, so for this question, the first thing that we need to do is we need to determine the amount of air within this house, right? so we can do that by the volume of air is equal to, so let's find the volume first.

00:19 So we have the area is 275 meters square, so volume is linked times width times height, right? so 275 meter square is the area that's already the link time width.

00:30 So now we have to multiply it by the height and they tell us it's 2 .5 meters right so this if we plug this into the calculator this will give us volume but let's convert this meters cube into liters right so i've written the conversion of meters cube to liters down here so one meters cube is equal to a thousand liters so we get the volume of error to be equal to 6 .88 times 10 to the 5 liters now the next thing we can do is we can convert the volume air to moles of air by using the density and the average molecular mass of the air.

01:13 So, moles, air is equal to 6 .88 times 10 to the 5 liters.

01:24 Right.

01:25 So they tell us the density of air is 1 .22 grams per liter.

01:31 And they also tell us the molar mass of air is 28 .9 grams per mo.

01:42 Right? so liters cancels out, grams cancels out, and we're left with moles of air.

01:48 And we get the answer to be 2 .9 times 10 to the 4 moles.

01:54 That is how many modes of air we have.

01:58 Now the next thing that we can do is determine how much heat, how much energy is needed to raise the temperature of air from 15 degrees celsius to 22 degrees celsius.

02:10 So i've converted the celsius into kelvin's.

02:12 All you've got to do is add 273, right? so it's 288 for the 15 degrees celsius and 295 for the 22 degrees celsius.

02:21 And the equation that we need to use to determine how much energy is needed is q is equal to m -c -delta -t, right? so q is equal to m.

02:33 So what is the m? so m we have 2 .9 times 10 to the 4 moes.

02:46 Now for c they tell us it's 29 .1 joules per mole.

02:52 So 29 .1 joules per mole.

02:58 Right, so this m can be in the units of moes or in the units of grams, depending on what the c is.

03:06 And since our c is jewels per moz, we leave it in most.

03:09 If it was joules per grams, then we would need to convert this moles into grams.

03:15 Okay, so, oops, now for the delta t.

03:20 Delta t is t final minus t initial, right? so 295 kelvin minus 288 kelvin...

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