To find the mass of methane gas produced, we first need to determine the molar ratio of CH3OH to CH4 using the balanced chemical equation for the reaction:

CH3OH -> CH4 + H2O

From the balanced equation, we can see that 1 mol of CH3OH produces 1 mol of CH4.

Next, we need to convert the given enthalpy change from kJ to J by multiplying by 1000:

82.4 kJ = 82,400 J

Now, we can use the equation:

ΔH = q = mcΔT

Where:
ΔH = enthalpy change (82,400 J)
m = mass of CH4 produced (in grams)
c = specific heat capacity of CH4 (assuming constant pressure)
ΔT = change in temperature (assuming constant pressure)

Since the reaction is assumed to occur at constant pressure, we can simplify the equation to:

ΔH = q = mcΔT

Now, we need to find the molar mass of CH4:

Molar mass of CH4 = 12.01 g/mol + 4(1.01 g/mol) = 16.05 g/mol

Now, we can use the molar ratio and molar mass of CH4 to determine the mass of methane gas produced:

1 mol CH3OH produces 1 mol CH4
Molar mass of CH4 = 16.05 g/mol

Therefore, 82,400 J of heat produced equals the production of 16.05 g of methane gas.

So, the mass of methane gas produced is 16.05 g.

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