Final answer:
To calculate the mass of methane required, we need to know the molar mass of methane, the heat required to increase the temperature of the air, and the heat produced by 1 mole of methane. Once we know these values, we can use the equations Q = mcΔT and Q = nΔH to calculate the number of moles of methane required. Then, we can calculate the mass of methane required using the equation Mass = n * molar mass. Approximately 0.1507 grams of methane is required to increase the temperature of the air in the dorm room by 8.38°C.
Step-by-step explanation:
The mass of methane required to increase the temperature of the air in the dorm room can be calculated using the equation for the combustion of methane: CH4(g) + 2O2(g) → CO2(g) + 2H2O(l) + 890.4 kJ/mol. From this equation, we can see that 1 mole of methane produces 890.4 kJ of heat. To calculate the mass of methane required, we need to know the molar mass of methane, which is 16 g/mol. To increase the temperature of the air by 8.38°C, we need to calculate the amount of heat required, and then use the heat produced by 1 mole of methane to find the mass.
First, we calculate the heat required using the equation Q = mcΔT, where Q is the heat required, m is the mass of air, c is the specific heat capacity of air, and ΔT is the change in temperature. Next, we use the equation Q = nΔH, where Q is the heat produced by 1 mole of methane, n is the number of moles of methane, and ΔH is the heat produced by 1 mole of methane.
Finally, we rearrange the equation to solve for n (the number of moles of methane):
n = Q / ΔH
Once we know the number of moles of methane, we can use the molar mass of methane to calculate the mass of methane required:
Mass = n * molar mass
Now we can substitute the values into the equations to calculate the mass of methane required to increase the temperature of the air in the dorm room by 8.38°C.
Let's assume the mass of air is 1000 g, the specific heat capacity of air is 1 J/g°C, and the ΔT is 8.38°C.
Using the equation Q = mcΔT, we get:
Q = 1000 g * 1 J/g°C * 8.38°C = 8380 J
Next, we calculate the heat produced by 1 mole of methane. From the balanced equation, we know that 1 mole of methane produces 890.4 kJ of heat. Converting kJ to J, we get:
ΔH = 890.4 kJ * 1000 J/kJ = 890400 J
Now we can substitute the values into n = Q / ΔH:
n = 8380 J / 890400 J/mol = 0.009416 mol
Finally, we calculate the mass of methane required using the equation Mass = n * molar mass:
Mass = 0.009416 mol * 16 g/mol = 0.1507 g
Therefore, approximately 0.1507 grams of methane is required to increase the temperature of the air in the dorm room by 8.38°C.