A sample comprising 0.0100 mol ethane (C2H6) is combusted in excess oxygen inside a bomb calorimeter that has a calorimeter constant of 440 J K–1. The calorimeter temperature increases from 22.0 to 26.6 °C. What is the change in internal energy accompanying this reaction?

Question

BlackTom AI · Accepted Answer

We start by identifying the given data and what is being asked. A bomb calorimeter with calorimeter constant C = 440 J K–1 is used, and the temperature rises from 22.0 °C to 26.6 °C, so ΔT = 26.6 − 22.0 = 4.6 °C. The sample combusted is 0.0100 mol of ethane (C2H6).

First, determine the heat exchanged with the calorimeter: q_cal = C × ΔT = 440 J K–1 × 4.6 K = 2024 J. This is the heat absorbed by the calorimeter.

In a bomb calorimeter, constant volume conditions apply, so the heat released by the reaction equals the decrease in the internal energy of the reacting system: ΔU = −q_reaction. Since q_cal is the heat gained by the calorimeter, the heat released by the reaction is q_reaction = +q_cal, thus ΔU = −q_cal = −2024 J for the 0.0100 mol of ethane.

To express this per mole, divide by the number of moles combusted: ΔU per mole = (−2024 J) / (0.0100 mol) = −202,400 J mol–1 = −202 kJ mol–1. This indicates that the internal energy of the system decreases by about 202 kJ per mole of ethane combusted.

Now evaluate the given options:
- Option a: a. 2020 kJ mol–1 — This value is far too large by about three orders of magnitude. It also uses a positive sign, which would imply energy is gained rather than released, contradicting the exothermic nature of combustion.
- Option b: b. 20.2 kJ mol–1 — This is an order of magnitude too small. It would correspond to a heat release about 100 times smaller per mole than the correct calculation, indicating a misapplication of the calorimeter reading or an incorrect division by moles.
- Option c: c. 202 kJ mol–1 — This matches the magnitude computed by dividing the total heat (2024 J) by the moles (0.0100 mol) and rounding appropriately. It reflects the correct per-mole internal-energy change in magnitude, though note the sign is negative for an exothermic process.
- Option d: d. 0.202 kJ mol–1 — This is far too small by three orders of magnitude; it would correspond to 0.202 kJ per 1 mol, which contradicts the actual energy scale observed from the calorimeter data.

To connect the reasoning: the measured temperature rise directly gives the heat change in the calorimeter, which for a bomb calorimeter equals the negative of the internal energy change of the reacting system. Converting to per-mole terms clarifies why the value around 202 kJ mol–1 appears as the correct magnitude for ΔU, consistent with the data provided and standard calorimetry relations. The other options stem from miscalculations of either the heat, the molar basis, or the sign, leading to values that are either off by factors of 10, 100, or lack the expected negative sign for an exothermic process.

A sample comprising 0.0100 mol ethane (C2H6) is combusted in excess oxygen inside a bomb calorimeter that has a calorimeter constant of 440 J K–1. The calorimeter temperature increases from 22.0 to 26.6 °C. What is the change in internal energy accompanying this reaction?

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