Chemistry · General chemistry I · Concept

Calorimetry and specific heat (q = mcΔT)

Calorimetry measures heat through a temperature change. For a sample that stays in one phase, q = mcΔT, where c is the specific heat. In an insulated calorimeter the heat lost by one part is gained by the rest, so one measured temperature change gives an unknown specific heat or a reaction’s enthalpy.

q = mcΔT

Heat q is energy that flows because of a temperature difference. For a sample that stays in one phase, the heat it absorbs is proportional to its mass m and its temperature change ΔT. The constant c, the specific heat, is the heat that warms 1 g by 1 °C. A change of 1 °C is the same size as a change of 1 K, so ΔT can be in either unit.

q = m c Δ T, Δ T = T_{final} - T_{initial}

\begin{array}{c} q = m c Δ T \\ Δ T = T_{final} - T_{initial} \end{array}

The sign says which way heat flows

ΔT is final minus initial. A sample that warms has ΔT > 0 and q > 0: it absorbs heat. A sample that cools has q < 0: it releases heat. Keep the sign, because the heat balance below depends on it.

Specific heat and heat capacity

Specific heat belongs to a substance and is measured per gram, in J/(g·°C). Heat capacity, C = mc, belongs to a whole object, such as a calorimeter, in J/°C. Water’s specific heat, 4.184 J/(g·°C), is high: the metal in the first worked example needs only about a tenth as much heat per gram for each degree.

Heat exchange in an insulated calorimeter

In an insulated container no heat escapes, so the heat released by the hot parts is absorbed by the cold parts: their heats add to zero. That one equation gives one unknown, such as a specific heat or the final temperature. A calorimeter that absorbs heat itself enters the sum through its heat capacity.

q_{hot} + q_{cold} = 0

Coffee-cup calorimetry measures ΔH

A reaction in solution at constant pressure releases or absorbs heat, and the solution takes it up or supplies it: q_rxn = −q_soln. Dividing by the moles that react gives ΔH per mole. For dilute aqueous solutions the solution is usually treated as water, with density 1.00 g/mL and specific heat 4.184 J/(g·°C).

q_{rxn} = - q_{soln}, Δ H = \frac{q_{rxn}}{n}

\begin{array}{c} q_{rxn} = - q_{soln} \\ Δ H = \frac{q_{rxn}}{n} \end{array}

Bomb calorimetry works at constant volume

A bomb calorimeter burns a sample inside a sealed steel vessel surrounded by water. Its heat capacity C_cal is found by calibration, so q_rxn = −C_cal ΔT. Because the volume is fixed, the result is the internal energy change ΔE rather than ΔH.

q_{rxn} = - C_{cal} Δ T

Phase changes and heating curves

While ice melts or water boils, added heat changes the phase and the temperature stays constant, so q = mcΔT does not apply. A phase change takes q = mL, where L is the heat per gram for melting or boiling. A heating curve alternates sloped segments, which use q = mcΔT, with flat ones, which use q = mL.

q = m L

Common mistakes

Using q = mcΔT across a melting or boiling point: during a phase change the temperature stays constant while heat flows.
Dropping a sign: a sample that cools has q < 0, and the heat balance q_hot + q_cold = 0 works only with the signs kept.
Confusing specific heat, which is per gram, with heat capacity, which is for a whole object.
Using only the mass of a dissolved solid for m in coffee-cup calorimetry; the whole solution warms or cools.
Reporting q_soln as the reaction’s ΔH: the reaction’s heat has the opposite sign and is divided by the moles that react.

Key terms

Calorimetry: Measuring the heat of a reaction or process from the temperature change it causes in a known material, using q = mcΔT. The calorimeter itself can absorb some heat, which careful work includes.
Specific heat capacity: The heat needed to raise the temperature of 1 g of a substance by 1 °C (or 1 K). Water’s is 4.184 J/(g·°C), and the heat is q = mcΔT.
Heat capacity: The heat needed to raise the temperature of a whole object or system by 1 °C (or 1 K), in J/°C. Specific heat is the same idea per gram.
Latent heat: The heat absorbed or released during a phase change, such as melting or boiling, while the temperature stays constant.
Heating curve: A graph of a sample’s temperature against the heat added. Sloped segments warm one phase (q = mcΔT); flat segments are phase changes, where the added heat changes the phase at constant temperature (q = mL).
Bomb calorimeter: A sealed steel container for burning a sample at constant volume. The heat it measures gives the reaction’s change in internal energy, ΔE, rather than ΔH.
Thermochemistry: The study of the energy changes, especially heat flow, that accompany chemical reactions and physical changes.

Work through an example

A 150.0 g metal block at 100.0 °C is dropped into 100.0 g of water at 20.0 °C in an insulated cup, and both end at 31.1 °C. What is the metal’s specific heat? Ignore the heat taken up by the cup.

Find the specific heat of a metal →

Find ΔH with a coffee-cup calorimeter →

Calculate the heat along a heating curve →

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