You want to heat your house with natural gas CH4. The house has 275 sq. m of the floor area and the height of 2.5 m (between the floor and the ceiling). The air has a specific molar heat capacity of 29.1 j/mol*K. The molar mass of air is 28.9 and a density of 1.22 g/L at these temperatures. How much methane do you have to use in order to heat air from 15 to 22 oC in your house?
I changed the temperatures to Kelvin, so they are now 288K and 295K. I also found the volume by multiplying 275 by 2.5 = 687.5. And that means that the mass of air is 838.75. After that Im stuck.
Do I use PV=nRT? or q = s * m * change in t?
Also, am I looking for grams of methane? Or volume? Or mass?
And, the problem only says "air"... Does that mean O2?
Im sorry for all the questions! Thank you for the help though!
q = mass air x sp. h. air x delta T will give you the heat required in J. Since sp. h. of air is given in J/mol*K, you need to convert mass of air to mols air using the mass/mol of 28.9 g/mol. What's missing from the problem is the heat generated by the combustion of CH4 and you need that before you can get grams of CH4. Perhaps its in the problem or you can look it up somewhere. As for the O2 or air, you are heating air but you are burning O2. My take on the problem is that the combustion of CH4 does not depend upon the air in the room but the air for the combustion is coming from another source. So whether it's oxygen or air mixed with oxygen need not enter into the problem. I would think either volume or grams would do for the amount of CH4. I would go with grams if I had a choice but I don't have a reason for that choice.
I found on another website that the heat of the reaction of CH4 is 50.1 kj/kg. Can this number be used in my question as well? I don't know where else to find this information.
After I get this information what do I need to do? Which formula is needed?
Thank you for your help so far!
heat needed is mass air x sp. h. air x delta T. You have the MOLAR specific heat of air so you need to change the volume to grams to mols. Calculate q.
CH4 + 2O2 ==> CO2 + 2H2O + 50.1 kJ/kg heat generated.
Knowing q (in J/g) and the reaction produces 50.1 kJ/kg, calculate grams CH4 needed. (By the way, since delta T is a difference, it makes no difference if you subtract C from C or K from K, the difference is 7 degrees.)
No problem, let's break down the problem step by step.
First, let's address your calculation for the mass of air. The volume of the house is given by the area multiplied by the height, so it should be 275 sq. m * 2.5 m = 687.5 cubic meters. The density of air is given as 1.22 g/L, which means it is 1.22 g per 1000 cubic cm (since 1 L = 1000 cubic cm). Therefore, the mass of air in the house is 687.5 cubic meters * 1.22 g / 1000 cubic cm = 0.83875 kg or 838.75 g.
Now let's move on to the heat equation. We can use the equation q = s * m * ΔT, where q is the amount of heat, s is the specific heat capacity, m is the mass, and ΔT is the change in temperature.
In this case, we're looking for the amount of methane (CH4) needed to heat the air, so we need to find the heat required to raise the temperature of air from 15°C to 22°C.
The change in temperature, ΔT, is 22°C - 15°C = 7°C. But remember, we need to convert this to Kelvin, so ΔT = 7 K.
The specific molar heat capacity of air is given as 29.1 J/mol*K, which means it takes 29.1 J to raise the temperature of 1 mole of air by 1 K.
To calculate the amount of heat required to heat the air in the house, we need to convert the mass of air to moles. The molar mass of air is given as 28.9 g/mol. So, the number of moles of air is 838.75 g / 28.9 g/mol = 28.971 mol.
Now we can calculate the amount of heat required using the equation:
q = s * m * ΔT
= 29.1 J/mol*K * 28.971 mol * 7 K
Calculating this:
q = 29.1 J/mol*K * 28.971 mol * 7 K
= 5925.427 J
The amount of heat required to heat the air is 5925.427 J.
Now, to determine the amount of methane (CH4) needed, we need to consider the energy content of CH4. The heat of combustion of methane is approximately 802 kJ/mol.
To find the amount of methane required, we can set up a ratio:
802 kJ/mol CH4 = 5925.427 J / x mol CH4
Solving for x (moles of CH4):
x = (5925.427 J * 1 mol CH4) / (802 kJ)
= 7.39 mol CH4
Therefore, you would need approximately 7.39 moles of methane (CH4) to heat the air in your house from 15 to 22°C.
Note: The problem only mentions "air," so we assume it refers to the mixture that makes up air, which includes nitrogen (N2), oxygen (O2), and other gases along with trace amounts of methane (CH4).