Heat of Reaction & Calorimetry — Theory
Calorimetry measures the heat exchanged when a reaction or physical process occurs.
We treat the reacting mixture as the system and the calorimeter assembly
(solution + container + bomb, if used) as the surroundings. Energy
conservation gives
\[
q_{\mathrm{system}} + q_{\mathrm{surroundings}} = 0
\quad\Longrightarrow\quad
q_{\mathrm{rxn}} = -\,q_{\mathrm{cal}}
\]
Sign convention: if the surroundings warm up (\(\Delta T>0\)), heat was released by the
reaction, so \(q_{\mathrm{rxn}}<0\) (exothermic). If the surroundings cool, \(q_{\mathrm{rxn}}>0\)
(endothermic).
Coffee-cup (constant-pressure) calorimetry
In an open cup the pressure is (approximately) constant, so the measured heat equals the
enthalpy change for the observed process:
\[
q_p = \Delta H \quad \text{(for the mixture observed)}
\]
The surroundings usually include the solution and the cup (and sometimes a stirrer/thermometer).
The heat they gain is
\[
q_{\mathrm{soln}} = m_{\mathrm{soln}}\,c_p\,\Delta T,
\qquad
q_{\mathrm{cup}} = C_{\mathrm{cal}}\,\Delta T,
\]
where \(m_{\mathrm{soln}}\) is the total mass of the solution,
\(c_p\) its specific heat capacity (often taken as water’s,
\(4.184\ \mathrm{J\,g^{-1}}\,^{\circ}\mathrm{C}^{-1}\)), and \(C_{\mathrm{cal}}\) the
(optional) heat capacity of the cup. The reaction heat is then
\[
q_{\mathrm{rxn}} = -\bigl(q_{\mathrm{soln}} + q_{\mathrm{cup}}\bigr)
= -\Bigl(m_{\mathrm{soln}}\,c_p + C_{\mathrm{cal}}\Bigr)\,\Delta T .
\]
For strong acid–base neutralization at room temperature, reporting \(\Delta H\) “per mole of
water formed” is customary. The moles of water are the limiting moles of \(\mathrm{H^+}\) or
\(\mathrm{OH^-}\).
Bomb (constant-volume) calorimetry
In a sealed steel bomb the volume is constant. The calorimeter’s temperature change gives
\[
q_{\mathrm{cal}} = C_{\mathrm{cal}}\,\Delta T,
\qquad
q_{\mathrm{rxn}} = -\,q_{\mathrm{cal}} .
\]
At constant volume the measured heat equals the change in internal energy
(\(q_v=\Delta U\)). For reactions with gases, the enthalpy is related by
\[
\Delta H = \Delta U + \Delta n_{\mathrm{gas}}\,R\,T ,
\]
where \(\Delta n_{\mathrm{gas}}\) is the change in moles of gas in the chemical equation.
Many introductory problems report the result simply as “heat of combustion” per gram or per mole:
\(\,q_{\mathrm{rxn}}/m\) or \(q_{\mathrm{rxn}}/n\).
From heat to molar quantities
To compare reactions, divide by the amount of the reaction basis:
\[
\Delta H_{\mathrm{rxn}}\ \text{(per mole)} \;=\;
\frac{q_{\mathrm{rxn}}}{n_{\text{limiting}}} \quad
\text{(cup; constant \(p\))}, \qquad
\frac{q_{\mathrm{rxn}}}{n} \quad \text{(bomb; per mole sample)} .
\]
For neutralization, the basis is moles of \(\mathrm{H_2O}\) formed (minimum of
\(n(\mathrm{H^+})\) and \(n(\mathrm{OH^-})\)). For combustion, the basis is usually
one mole of fuel.
Units & practical notes
- Temperature change: \(\Delta T = T_f - T_i\). A difference has the same magnitude in
\(^{\circ}\mathrm{C}\) and \(\mathrm{K}\).
- Typical assumptions in cup calorimetry: density \(\approx 1.00\ \mathrm{g\,mL^{-1}}\),
solution \(c_p \approx 4.184\ \mathrm{J\,g^{-1}}\,^{\circ}\mathrm{C}^{-1}\), negligible heat loss
to air (or corrected by an empirical \(C_{\mathrm{cal}}\)).
- Bomb calorimeters are first calibrated by burning a standard (e.g., benzoic acid)
to determine \(C_{\mathrm{cal}}\).
- Report signs: exothermic \(q_{\mathrm{rxn}}<0\) (temperature rises), endothermic
\(q_{\mathrm{rxn}}>0\) (temperature falls).
Common pitfalls
- Mixing “per gram” and “per mole” bases; always state the basis clearly.
- For bomb data, quoting \(\Delta H\) without correcting from \(\Delta U\) when large
gas-mole changes are present.
- For cup data, forgetting the cup/stirrer heat capacity or using volumes instead of mass
without applying density.
At-a-glance equations used by this calculator
\[
\boxed{\,q_{\mathrm{cal}} = C_{\mathrm{cal}}\,\Delta T\,}
\qquad
\boxed{\,q_{\mathrm{rxn}} = -\,q_{\mathrm{cal}}\,}
\]
\[
\boxed{\,q_{\mathrm{soln}} = m_{\mathrm{soln}}\,c_p\,\Delta T,\quad
q_{\mathrm{rxn}} = -\bigl(q_{\mathrm{soln}} + q_{\mathrm{cup}}\bigr)\,}
\]
Enter any two of \(T_i\), \(T_f\), and \(\Delta T\); choose bomb or cup; and (optionally)
provide masses/moles to obtain per-gram or per-mole quantities.
