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To generalize the problem, I will just say:

5 gram of impure ethanol (or any other type of combustible material) is placed inside a bomb calorimeter where volume is constant. If the combustion produces a certain amount of energy, how should I find the mass in gram of pure ethanol contained in that quantity of impure ethanol?

Here is what I will do:

  • find the standard molar enthalpy of ethanol.
  • write a combustion equation.
  • find the energy produced in the combustion of one mole of ethanol.
  • then divide that amount of ethanol by the amount of energy stated in problem.

To find how many mole of pure ethanol, convert the mole into gram.

But there are two things that make me think that this way of solving the problem is not correct:

  • the quantity of impure ethanol is a mixture of pure ethanol and another unknown material. What if this unknown material produce part of the energy stated in the problem?
  • the standard molar enthalpy is measure under standard conditions (1 atm, 25 °C). However, the condition inside the bomb calorimeter is unknown.
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What if this unknown material produce part of the energy stated in the problem?

It would be stated in the problem that the amount of heat that the unknown material produced.

The standard molar enthalpy is measured under standard conditions (1 atm, 25°C). However, the condition inside the bomb calorimeter is unknown.

We are given some information, the final temperature of the system. But you are right in that we do not know the pressure, and the specific heat of the remaining items is difficult to calculate.

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That is impossible. If your compound is not pure and you do not know the amount of impurity and what kind of impurity, you cannot possibly calculate the amount of ethanol present. Unless we assume that the impurity is small or incombustible.

I wouldn't use a bomb calorimeter in this case because you want to work with enthalpy, not internal energy. An adiabatic flame calorimeter would be more suitable. If you need to use a bomb calorimeter, you can convert between $\Delta U$ and $\Delta H$ by tracking the pressure in the bomb. You can of course control the conditions inside the bomb at the start of the experiment.

The standard ethalpy of combustion can still be used, provided that the end temperature of the products of the chemical of physical process are not much higher than the starting temperature of the reactants. A large enough reservoir of heat absorbent or higher $C_v$ (calorimeter constant) should ensure this, but not too large or too high, else you won't be able to accurately read of the temperature difference (you see what the advantage is of an accurate thermometer in this case).

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