Combustion of natural gas (primarily methane) occurs in most household heaters. The heat given off in this reaction is used to raise the temperature of the air in the house. Assuming that all the energy given off in the reaction goes to heating up only the air in the house, determine the mass of methane required to heat the air in a house by 10.0 degrees C. Assume each of the following: house dimensions are 30.0 m x 30.0 m x 3.0 m; specific heat capacity of air is 30 J/k•mol; 1.00 mol of air occupies 22.4 L for all temperatures concerned.
My work:
2700 m^3 = 12 mol of air
q= 12 mol x 30 J/k•mol x 283.15 K
q= 1.0 x 10^5 J
Am I on the right track?
I got -890.3 KJ for the change in enthalpy heat for the balanced equation:
CH4 + 2O2 ---> 2H2O + CO2
From then on I would multiply ...
q x 1kJ/1000 J x 1mol/890.3 x 16.05 g CH4/1 mol CH4
However, my answer is incorrect based on the answer key.
how in the world did you compute 12 moles of air occupies 2700m^3?
1 mole of gas (any gas) occupies 22.4dm^3
You are on the right track, but there seems to be a slight error in your calculations. Let's go through the steps again to determine the mass of methane required to heat the air in the house.
First, we need to calculate the amount of heat required to raise the temperature of the air in the house by 10.0 degrees Celsius. You correctly determined that the volume of the house is 2700 m^3, which corresponds to 12 moles of air (using 1.00 mol of air occupies 22.4 L).
Next, we can use the specific heat capacity of air (30 J/k•mol) to calculate the heat required:
q = 12 mol x 30 J/k•mol x 10.0 K
q = 3,600 J
So, the amount of heat required to raise the temperature of the air in the house by 10.0 degrees Celsius is 3,600 J.
Now, let's calculate the mass of methane required to release this amount of heat.
From the balanced equation of the combustion reaction:
CH4 + 2O2 -> 2H2O + CO2
We know that the change in enthalpy for this reaction is -890.3 kJ (negative because heat is released). So, we can convert this to joules:
ΔH = -890.3 kJ x 1000 J/kJ
ΔH = -890,300 J
Now, using the stoichiometry of the reaction, we can determine the amount of methane required to release this amount of heat. From the balanced equation, we see that we need 1 mole of methane to release -890.3 kJ of heat.
Therefore, for 3,600 J of heat, we would need:
(3,600 J / -890,300 J) x 1 mol CH4 = 0.00404 mol CH4
Finally, to convert moles of methane to grams, we multiply by the molar mass of methane (16.04 g/mol):
0.00404 mol CH4 x 16.04 g/mol = 0.0649 g CH4
So, the mass of methane required to heat the air in the house by 10.0 degrees Celsius is approximately 0.0649 grams.
Your approach is mostly correct, but there seems to be a mistake in the conversion factor used to convert from moles of methane to grams of methane. Let's go through the calculations step-by-step to determine the correct mass of methane required.
1. Calculate the volume of air in the house:
Volume of house = length x width x height = 30.0 m x 30.0 m x 3.0 m = 2,700 m^3
2. Convert the volume of air to moles:
1.00 mol of air occupies 22.4 L
So, 2,700 m^3 of air = (2,700 x 1,000 L) / 22.4 L/mol = 120,536.6 mol of air
3. Calculate the heat required to raise the temperature of the air:
q = mass x specific heat capacity x change in temperature
q = 120,536.6 mol x 30 J/(K•mol) x 10.0 °C = 36,160,980 J
Now, let's calculate the mass of methane required:
4. Calculate the moles of methane required using the balanced equation:
CH4 + 2O2 -> 2H2O + CO2
From the balanced equation, we can see that 1 mole of methane produces 2 moles of water.
So, moles of methane = 120,536.6 mol of water / 2 = 60,268.3 mol of methane
5. Calculate the heat of combustion for 1 mole of methane:
From the given ΔH° values:
ΔH° = -890.3 kJ/mol
Convert to J/mol: -890.3 kJ x 1,000 J/1 kJ = -890,300 J/mol
6. Calculate the heat required to heat the air using the heat of combustion of methane:
36,160,980 J of air / 890,300 J/mol = 40.58 mol of methane
7. Convert the moles of methane to grams:
40.58 mol x 16.04 g/mol = 651.9 g of methane
Therefore, the mass of methane required to heat the air in the house by 10.0 °C is approximately 651.9 grams.