Questions: Heat Capacity and Calorimetry

Question 1

Two 100 g samples are each supplied with 1,000 J of heat: one sample is water (c = 4.184 J/g·°C) and one is iron (c = 0.449 J/g·°C). Which sample reaches a higher final temperature, and why?

• A. Water, because its high specific heat allows it to store more energy per gram
• B. Iron, because its low specific heat means the same energy input produces a much larger temperature increase
• C. Both reach the same final temperature because they received identical amounts of energy
• D. Iron, because metals conduct heat more efficiently than liquids

Answer: (B)

From q = mcΔT, rearranging gives ΔT = q/(mc). With identical q and m, the sample with the smaller specific heat c shows a larger temperature increase. Iron's specific heat is about nine times smaller than water's, so iron's temperature rises roughly nine times more for the same energy input. This is why a metal pan handle becomes dangerously hot while the water inside barely warms — and why water is such an effective coolant. Option 3 confuses thermal conductivity (how fast heat moves through a material) with specific heat (how much energy is needed per degree of temperature change).

Question 2

A chemist burns a sample inside a sealed, rigid steel vessel submerged in a water bath and measures a temperature rise of 3.5°C. The calorimeter constant is 5.0 kJ/°C. What thermodynamic quantity does this experiment directly measure?

• A. ΔH (enthalpy change at constant pressure)
• B. ΔG (Gibbs free energy change)
• C. ΔE (internal energy change at constant volume)
• D. The activation energy of the combustion reaction

Answer: (C)

The sealed, rigid vessel means the volume cannot change, so no pressure-volume work is done on or by the surroundings. Under constant-volume conditions, the heat flow equals the change in internal energy (ΔE), not enthalpy (ΔH). This is why bomb calorimeters measure ΔE directly. The enthalpy can be calculated afterward using ΔH = ΔE + ΔnRT, where Δn is the moles of gas produced minus consumed. For combustion reactions with significant gas moles, the correction can be non-trivial. The coffee-cup calorimeter (constant pressure, open to atmosphere) measures ΔH directly.

Question 3

A large pot of water at 30°C contains more total thermal energy than a small thimble of water at 90°C.

• T. True
• F. False

Answer: True

This illustrates the critical distinction between temperature and heat. Temperature measures the average kinetic energy per molecule — the thimble's molecules are more energetic on average. But total thermal energy depends on both average energy per molecule AND the number of molecules. The large pot has vastly more water molecules, so the total energy stored (q = mcΔT, relative to some reference) can far exceed the thimble's total. This is why the ocean, even at a relatively cool 15°C, stores an enormous amount of thermal energy — enough to significantly moderate coastal climates.

Question 4

In a well-insulated coffee-cup calorimeter, if the solution temperature rises after two reactants are mixed, the reaction that occurred was endothermic.

• T. True
• F. False

Answer: False

A temperature rise in the calorimeter solution means heat was released by the reaction into the solution — this is an exothermic reaction (ΔH < 0). In calorimetry, q(reaction) = −q(solution): heat lost by the reaction equals heat gained by the solution. If the solution temperature goes up, the solution absorbed heat, meaning the reaction released heat — exothermic. An endothermic reaction would cool the solution as it draws heat from the surroundings. This sign-convention confusion is one of the most common errors in calorimetry problems.

Question 5

A bomb calorimeter and a coffee-cup calorimeter are used to measure the heat of combustion of the same compound. Explain why they give slightly different numerical results, and identify which thermodynamic quantity each one directly measures.

Model answer: A coffee-cup calorimeter operates at constant pressure (open to the atmosphere), so the heat measured directly equals the enthalpy change ΔH. A bomb calorimeter operates at constant volume (sealed rigid vessel), so the heat measured directly equals the internal energy change ΔE. The two quantities are related by ΔH = ΔE + ΔnRT, where Δn is the change in moles of gas during the reaction. For combustion reactions that produce or consume gas, Δn ≠ 0, so ΔH and ΔE differ by a small but non-negligible amount. For reactions with no change in gas moles, the two are essentially equal.

The conceptual key is that enthalpy accounts for the energy of expansion against atmospheric pressure (the PV work term), while internal energy does not. In a sealed bomb, no expansion work is possible — all the energy shows up as heat. In an open coffee cup, if the reaction produces gas, some energy goes into pushing back the atmosphere, which is why ΔH (which includes this work) differs from ΔE. For most aqueous reactions with no gas involvement, the difference is negligible.