First, we need to determine the mass of water in the pool:
mass = density x volume
density of water = 1000 kg/m³
volume = length x width x depth
volume = 10.0 m x 4.0 m x 3.0 m = 120 m³
mass = 1000 kg/m³ x 120 m³ = 120000 kg
Next, we need to calculate the heat required to raise the temperature of the water:
q = m x c x ΔT
where q is the heat energy, m is the mass of water, c is the specific heat of water, and ΔT is the change in temperature.
c = 4.18 J/g°C (specific heat of water)
ΔT = 27.3°C - 20.2°C = 7.1°C
m = 120000 kg
q = 120000 kg x 4.18 J/g°C x 7.1°C = 35792400 J
Next, we need to convert the energy required to burn methane to heat energy:
-891 kJ/mol x (1 mol CH4/160 g CH4) x (1000 g/1 kg) = -5.569 kJ/g
We can now calculate the amount of methane needed:
energy = -5.569 kJ/g x mass CH4
mass CH4 = energy / (-5.569 kJ/g)
mass CH4 = 35792400 J / (-5569 J/g) = -6431.6 g
At STP, 1 mole of any gas occupies 22.4 L of volume. We can use this to convert the mass of methane to volume at STP:
1 mol CH4 = 16 g CH4
-6431.6 g CH4 x (1 mol CH4/16 g CH4) x (22.4 L/1 mol CH4) = -9074.4 L
Since we cannot have a negative volume, we can take the absolute value of the result:
|9074.4 L| = 9074 L
Therefore, approximately 9074 liters of methane gas at STP must be burned to raise the temperature of the water in the pool from 20.2°C to 27.3°C.