Constant-Pressure and Bomb Calorimetry

Calorimetry is the measurement of the heat of a reaction or a process: a constant-pressure (coffee-cup) calorimeter measures the enthalpy change ΔH, and a constant-volume bomb calorimeter measures the internal energy change ΔU. In brief: mixing 50.0 mL of 1.00 M HCl with 50.0 mL of 1.00 M NaOH in a coffee-cup calorimeter raises the temperature by about 6.5 °C, which corresponds to ΔH ≈ −54 kJ per mole of water formed.

Calorimeters are the instruments by which most enthalpies of reaction and the energy contents of foods and fuels have been determined. They work by letting a reaction take place in a thermally isolated container and measuring the temperature change. This article explains how both types of calorimeter work, shows how to calculate ΔH from temperature data, how to find the calorimeter constant and how to convert between ΔU and ΔH.

What is the principle of calorimetry?

In a perfectly insulated calorimeter, the heat released or absorbed by the reaction is taken up or supplied by the contents of the calorimeter and the calorimeter itself:

qreaction = −(qsolution + qcalorimeter)

where q_solution = m × c × ΔT and q_calorimeter = C_cal × ΔT. The temperature of the calorimeter rises if the reaction is exothermic (q_reaction < 0) and falls if it is endothermic. To find the heat per mole, q is divided by the number of moles of the limiting reagent.

How does a constant-pressure calorimeter work?

The simplest version is a coffee-cup calorimeter: two nested polystyrene cups with a lid, a thermometer and a stirrer. The reaction takes place in solution, open to the atmosphere through a small hole, so the pressure is constant, and the heat measured is the enthalpy change: q_p = ΔH. It is used for reactions in dilute aqueous solution, such as neutralization, dissolving and simple precipitation. The solution is usually assumed to have the density (1.00 g/mL) and the specific heat (4.184 J/(g·°C)) of water, and the heat capacity of the cup is often small enough to neglect, or it is determined by a calibration.

How does a bomb calorimeter work?

A bomb calorimeter is used for combustion reactions. A weighed sample is placed in a strong steel container (the bomb) that is filled with oxygen at about 25–30 atm and sealed. The bomb is immersed in a known mass of water in an insulated vessel, and the sample is ignited electrically. The heat of combustion raises the temperature of the bomb and the water, and the temperature rise is measured to 0.001 °C.

Since the volume of the bomb is fixed, no pressure–volume work is done, so the heat is q_v = ΔU. The total heat capacity of the calorimeter (bomb plus water plus parts) is C_cal, found by burning a standard such as benzoic acid, which releases 26.43 kJ of heat per gram.

Comparison of the two types

Worked example: neutralization

50.0 mL of 1.00 M HCl and 50.0 mL of 1.00 M NaOH, both at 21.0 °C, are mixed in a coffee-cup calorimeter. The highest temperature reached is 27.5 °C. Find ΔH per mole of water formed. Assume that the density of the solution is 1.00 g/mL, c = 4.184 J/(g·°C), and the heat capacity of the cup is negligible.

1. Mass of solution = 100.0 mL × 1.00 g/mL = 100.0 g.
2. ΔT = 27.5 − 21.0 = 6.5 °C.
3. q_solution = 100.0 × 4.184 × 6.5 = 2720 J.
4. q_reaction = −2720 J.
5. Moles of water formed = 0.0500 L × 1.00 mol/L = 0.0500 mol.
6. ΔH = −2720 J / 0.0500 mol = −54,400 J/mol = −54 kJ/mol.

The accepted value for the neutralization of a strong acid with a strong base is about −56 kJ/mol. The difference comes from heat absorbed by the cup and lost to the air.

Worked example: combustion of glucose

A bomb calorimeter is calibrated by burning 1.000 g of benzoic acid, which releases 26.43 kJ. The temperature rises by 2.50 °C. Then 1.250 g of glucose (C₆H₁₂O₆, 180.16 g/mol) is burned and the temperature rises by 1.84 °C. Find the energy of combustion per mole.

1. C_cal = 26.43 kJ / 2.50 °C = 10.57 kJ/°C.
2. q = C_cal × ΔT = 10.57 × 1.84 = 19.45 kJ released by glucose.
3. n(glucose) = 1.250 / 180.16 = 0.006938 mol.
4. ΔU = −19.45 / 0.006938 = −2803 kJ/mol.

Converting ΔU to ΔH

For C₆H₁₂O₆(s) + 6O₂(g) → 6CO₂(g) + 6H₂O(l), Δn_gas = 6 − 6 = 0, so ΔH = ΔU = −2803 kJ/mol. For benzoic acid, C₇H₆O₂(s) + 7.5O₂(g) → 7CO₂(g) + 3H₂O(l), Δn_gas = 7 − 7.5 = −0.5, and ΔH = ΔU + Δn_gasRT = −3226 + (−0.5)(0.008314)(298) = −3227 kJ/mol. The difference is small for solids and liquids, because only the gases matter.

How did calorimetry develop?

Antoine Lavoisier and Pierre-Simon Laplace built an ice calorimeter in 1782–1783, in which the heat from a reaction or an animal was measured by the amount of ice that melted. Marcellin Berthelot developed the bomb calorimeter in the 1880s and used it to measure the heats of combustion of hundreds of organic compounds. Modern calorimeters use electronic temperature sensors and computer control, and the same principle is used to find the Calorie content of foods.

Which mistakes are common?

• Forgetting the sign: the heat gained by the solution is the opposite of the heat of the reaction.
• Using the mass of only one reactant solution: use the total mass.
• Neglecting the calorimeter constant: in a bomb calorimeter it is essential.
• Dividing by the wrong number of moles: use the limiting reagent.
• Mixing up ΔU and ΔH: ΔU is measured at constant volume, ΔH at constant pressure.