Skip to main content
Chemistry LibreTexts

7.2: Introduction to calorimetry

  • Page ID
    424819
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \( \newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\)

    ( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\id}{\mathrm{id}}\)

    \( \newcommand{\Span}{\mathrm{span}}\)

    \( \newcommand{\kernel}{\mathrm{null}\,}\)

    \( \newcommand{\range}{\mathrm{range}\,}\)

    \( \newcommand{\RealPart}{\mathrm{Re}}\)

    \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\)

    \( \newcommand{\Argument}{\mathrm{Arg}}\)

    \( \newcommand{\norm}[1]{\| #1 \|}\)

    \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\)

    \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    \( \newcommand{\vectorA}[1]{\vec{#1}}      % arrow\)

    \( \newcommand{\vectorAt}[1]{\vec{\text{#1}}}      % arrow\)

    \( \newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vectorC}[1]{\textbf{#1}} \)

    \( \newcommand{\vectorD}[1]{\overrightarrow{#1}} \)

    \( \newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}} \)

    \( \newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}} \)

    \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \)

    \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)

    \(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)
    Note

    There are two "traditional" calorimetry experiments that the you can choose from in this module (pending the availability of instruments). Both experiments apply the same fundamental theories of thermochemistry. But they differ in the type of thermochemical experiment:

    • Solution Calorimetry is a constant-pressure experiment
    • Bomb Calorimetry is a constant-volume experiment

    General Theory

    Definitions:

    Terms:

    • A system is the specific place where a reaction or process is occurring (eg inside calorimeter).
    • The surroundings are outside of the system (eg the lab bench, air around the calorimeter).
    • A closed system is one where no matter is exchanged to or from the surroundings. In contrast, an open system is one where material can enter or exit the system (like a car or the human body).
    • An isolated system is a closed system where no energy is exchanged to or from the surroundings.
    • Work (\(w\)) is the energy associated with moving matter or energy in a non-random direction (eg expanding a gas). \(w=\text{opposing force} \times \text{distance moved}\). When the force is pressure and the distance moved is related to volume, \(w=-PdV\)
    • Heat (\(q\)) is the energy transferred as a result of temperature differences. Heat is associated with rotational and translational movement in random directions.
    • Total energy (E) is the capacity to do work.
    • Internal energy (\(U\)) is the internal energy of a system.

    Symbols:

    • \(U =\) the internal energy of a system
    • \(q =\) heat
    • \(w =\) work
    • \(H=\) enthalpy
    • \(P =\) pressure
    • \(_p\) subscript means at constant pressure
    • \(V =\) volume
    • \(_v\) subscript means at constant volume
    • \(\Delta\) means the change in something (eg \(\Delta Y\) is change in \(Y\))
    • \(d\) means the differential (eg \(dH\) is the ordinary derivative of \(H\))
    • \(\partial\) means the partial derivative

    Energy

    Energy is absorbed or released during chemical changes. Thermochemical studies involve the measurement of that energy as heat absorbed or evolved during a chemical change. This permits quantification of the energy released or absorbed in the making or breaking of chemical bonds.

    The first law of thermodynamics states that energy is conserved. If energy enters a system, an equal amount must be lost by the system's environment. In the case of a closed system, the energy change for a system may be written as follows:

    \[\Delta {U}={q}+{w} \label{U}\]

    where \(\Delta U\) is the internal energy change of the system, \(q\) is the heat entering or leaving the system, and \(w\) is the energy change caused by work.

    If work is done on the system by an external source, this would result in an increase in the internal energy of the final state, thus \( w\) in Equation \ref{U} would be a positive quantity; if the system were to do work on its surroundings, \({w}\), would be negative. If \( q\) is positive, heat has entered the reaction system and the reaction is endothermic. Conversely, if \( {q}\) is negative, the reaction is exothermic.

    In chemical reactions, work is most often from changes in volume (V) due to a change in the moles of gas. Expansion work is expressed as \(w=-PdV\), where pressure (P) and volume (V). Thus, \ref{U} can be written:

    \[\Delta {U}={q}-PdV \label{U2}\]

    Constant volume

    In a constant volume experiment, no work is done, this \(\Delta U = q\).

    Let's take the combustion of glucose as an example. If glucose is combusted to form six molecules of carbon dioxide gas, the volume expands. Some of the energy released from the chemical reaction is heat, while some does expansion work. However, when a system has rigid walls (like a bomb calorimeter), and thus the volume does not change, no expansion work is done; thus all energy is transferred as heat, and \(\Delta U = q\).

    In this case, we can measure the energy gained or released by a chemical reaction by measuring heat; or temperature change. The heat is related to the measured temperature change (\(\Delta T\)) by the calorimeter by the calorimeter's heat capacity (\(C_{cal}\)):

    \[C_{cal} = \frac{q}{\Delta T}\]

    Where \(\Delta T = T_{\text{final}}-T_{\text{initial}}\)

    At constant volume, we can also express this as \(C_{v, cal} =\dfrac{\Delta U}{\Delta T}\), where the "v" subsript indicates constant voume.

    Provided that the \(C_{cal}\) is known, the heat \(q\), and thus energy of the reaction \(\Delta U\), can be determined by measuring \(\Delta T\) before and after the chemical reaction has occurred.

    Enthalpy

    The enthalpy (\(H\)) of a system is defined as \(H = U + PV \), where \(U\) is the internal energy of the system, \(P\) is pressure, and \(V\) is volume. We can also define a change in enthalpy as a change in each of the component state functions: \[\Delta H = \Delta U + P\Delta V\]

    Enthalpy at constant pressure

    Enthalpy lets us keep track of the energy of a system that is at constant pressure (like our solution calorimeters). And using \ref{U} we can write this expression as a function of heat and work:

    \[\Delta H = q + w + P \Delta V\]

    And we can substitute the expression \(w = -P\Delta V\) into this expression to show that \(\Delta H = q\):

    \[\Delta H = q -P \Delta V + P \Delta V = q\]

    Just as the case described above, we can determine \(q\) from a measurement of \(\Delta T\) using a known heat capacity for the calorimeter. At constant pressure, the calorimeter's heat capacity can be expressed as \(C_{p, cal} =\dfrac{\Delta H}{\Delta T}\), where the "p" subscript indicates constant pressure. Thus, it is clear that the change in enthalpy can be determined by measuring \(\Delta T\). However, this is not a direct measure of \(\Delta U\), because at constant pressure, some work may be done if there is a change in volume.

    Enthalpy involving gas

    When a gas is evolved as a product of the chemical reaction, the volume change, and the work done is significant. A relationship between \(\Delta H\) and \(\Delta U\) can be determined with some help from the ideal gas laws:

    \[\Delta H = \Delta U + \Delta n_{gas}RT \label{Hgas}\]

    where \(\Delta n_{gas}\) is the change in the moles of gas from reactants to products.


    This page titled 7.2: Introduction to calorimetry is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Kathryn Haas.

    • Was this article helpful?