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 answer: 649 g
Well, let's calculate it in a slightly different way, shall we?
First, let's find the volume of air in the house:
Volume = length x width x height
Volume = 30.0 m x 30.0 m x 3.0 m
Volume = 2700 m³
Now, let's convert this volume to liters because the given specific heat capacity is for 1.00 mol of air occupying 22.4 L:
Volume = 2700 m³ x 1000 L/m³
Volume = 2,700,000 L
Next, we need to find the number of moles of air in the house:
Number of moles = Volume / 22.4 L/mol
Number of moles = 2,700,000 L / 22.4 L/mol
Number of moles ≈ 120,536.61 mol
To raise the temperature of the air by 10.0 degrees C, we need to multiply the number of moles by the specific heat and the temperature change:
Energy = Number of moles x Specific heat x Temperature change
Energy ≈ 120,536.61 mol x 30 J/k•mol x 10.0 °C
Energy ≈ 36,160,983 J
Now, let's convert this energy to kilojoules:
Energy = 36,160,983 J / 1000
Energy ≈ 36,160.983 kJ
Finally, let's calculate the mass of methane required to produce this energy using the heat of combustion of methane (assuming it is 890 kJ/mol):
Mass = Energy / Heat of combustion
Mass ≈ 36,160.983 kJ / 890 kJ/mol
Mass ≈ 40.60 mol
And since the molar mass of methane (CH4) is approximately 16.04 g/mol, we can find the mass in grams:
Mass = 40.60 mol x 16.04 g/mol
Mass ≈ 651.64 g
So, it seems like you were pretty close with your answer, but the correct mass of methane required to heat the air in the house by 10.0 degrees C is approximately 651.64 grams.
To determine the mass of methane required to heat the air in a house by 10.0 degrees C, we need to calculate the amount of energy required to raise the temperature of the air and then convert it to the mass of methane.
Step 1: Calculate the volume of air in the house
The volume of the house can be calculated by multiplying its dimensions: 30.0 m x 30.0 m x 3.0 m = 2700 m^3.
Step 2: Convert the volume of air to moles
Since we know that 1.00 mol of air occupies 22.4 L at any temperature, we can calculate the number of moles of air in the house:
2700 m^3 x (1000 L / 1 m^3) / (22.4 L/mol) = 120535.71 mol
Step 3: Calculate the energy required to raise the temperature
The equation for calculating the energy required is: Q = mcΔT, where Q is the energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Since we want to raise the temperature of the air by 10.0 degrees C, ΔT = 10.0 degrees C.
Step 4: Calculate the mass of methane
To calculate the mass of methane required, we need to determine the energy given off per mole of methane during combustion.
The balanced combustion equation for methane (CH4) is:
CH4 + 2O2 → CO2 + 2H2O
From the equation, we know that 1 mole of methane produces 891 kJ of energy.
Using this information, we can calculate the mass of methane required:
Mass = (Q / ΔT) / (specific heat capacity x energy given off per mole) x molar mass of methane
Mass = (120535.71 mol x 30 J/k.mol x 10.0 degrees C) / (30 J/k.mol x 891 kJ/mol) x 16.04 g/mol
Mass = 649 g
Therefore, the mass of methane required to heat the air in the house by 10.0 degrees C is approximately 649 grams.
To determine the mass of methane required to heat the air in a house by 10.0 degrees Celsius, we need to follow these steps:
Step 1: Calculate the volume of air in the house.
The volume of the house can be calculated by multiplying its dimensions:
Volume = length × width × height
Volume = 30.0 m × 30.0 m × 3.0 m
Volume = 2700 m³
Step 2: Convert the volume of air to moles.
Since we are given the volume occupied by 1.00 mole of air at all temperatures, we can use the ideal gas law to convert the volume of air into moles.
1 mole of air = 22.4 L
So, to find the moles of air in the house:
Number of moles = Volume of air / 22.4 L
Number of moles = 2700 m³ / (22.4 L/m³)
Number of moles = 120.54 moles
Step 3: Calculate the heat energy required to raise the temperature.
The formula to calculate heat energy is:
Q = mcΔT
Where Q is the heat energy, m is the mass, c is the specific heat capacity, and ΔT is the change in temperature.
Assuming all the energy is used to heat the air, we can substitute the given values into the formula:
ΔT = 10.0°C
c = 30 J/k·mol
m is what we need to find.
Step 4: Rearrange the formula and solve for mass (m).
Q = mcΔT
m = Q / (c * ΔT)
m = 120.54 moles * 30 J/k·mol * 10.0°C
Calculating this, we find:
m = 36,162 J
Step 5: Convert the energy into grams by using the molar mass of methane (CH4).
The molar mass of methane (CH4) is 16.04 g/mol.
To convert from Joules to grams, we use the equation:
1 J = 1 kg·m²/s²
1 kg = 1000 g
1 m² = 1 m²
Therefore:
1 J = 1 kg·m²/s² = 1000 g·m²/s² = 1000 g/cm²·m²
Converting:
m = (36,162 J) / (1000 g/cm²·m²)
m = 36.162 g
Therefore, the mass of methane required to heat the air in the house by 10.0 degrees Celsius is approximately 36.162 grams.
CH4 + 2O2 ==> CO2 + H2O
dHrxn = (n*dHformation products) - (dHformation reactants) = ? = q for 1 mol CH4.
volume of house = 3000cm*3000cm*300cn = ? cc.
mols air = ?cc/22,400 = ?
How many Joules (or kJ) do you need to heat the house.
That's q needed = mols air x specific air (in J/mol) x delta T.
Then q for 1 mol CH4 x ?mols CH4 = qneeded.
Solve for number mols CH4 and convert to grams. Checl this against your work.