Methane burns in air at about 1957˚C. Suppose you make a Carnot engine which is fueled by methane at this temperature, and exhausts at 500 ˚C. Suppose you burn 1 kg of methane in this engine (at the burn temperature, the reaction products will be CO₂( g) and H₂O(g) ). How much work in Joules will the engine create?

Answer:

The Carnot engine operates based on the principles of the Carnot cycle, which is a theoretical idealized thermodynamic cycle. To calculate the work done by the engine, we need to use the formula for the efficiency of the Carnot cycle.

The efficiency of a Carnot engine is given by the equation:

Efficiency = 1 - (T2 / T1),

where T2 is the exhaust temperature in Kelvin and T1 is the burn temperature in Kelvin.

First, we need to convert the temperatures from Celsius to Kelvin.

The burn temperature is 1957 ˚C, so we add 273 to convert it to Kelvin:

T1 = 1957 + 273 = 2230 K.

The exhaust temperature is 500 ˚C, so we add 273 to convert it to Kelvin:

T2 = 500 + 273 = 773 K.

Now we can calculate the efficiency:

Efficiency = 1 - (T2 / T1) = 1 - (773 / 2230).

Next, we need to calculate the heat input, which is the energy released by burning 1 kg of methane.

The energy released by burning methane can be calculated using the heat of combustion of methane, which is -891 kJ/mol.

To convert this to joules per kilogram, we need to know the molar mass of methane, which is 16 g/mol.

1 kg of methane is equal to 1000 g, so the number of moles of methane in 1 kg is:

1000 g / 16 g/mol = 62.5 mol.

The heat released by burning 1 kg of methane is:

-891 kJ/mol * 62.5 mol = -55,687.5 kJ.

To convert this to joules, we multiply by 1000:

-55,687.5 kJ * 1000 = -55,687,500 J.

Now we can calculate the work done by the engine:

Work = Efficiency * Heat input.

Substituting the values we calculated:

Work = (1 - (773 / 2230)) * (-55,687,500 J).

Finally, we can calculate the work done by the engine in joules.

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