Q. how to calculate heat capacity \(Q\)

Definition. Heat capacity is the quantity that relates the amount of heat added to an object to the resulting change in temperature. The heat capacity of an object is denoted by \(C\) and defined by the formula

\[
C \;=\; \frac{Q}{\Delta T}
\]

In this formula, \(Q\) is the heat added to the object measured in \(\mathrm{J}\), and \(\Delta T\) is the change in temperature measured in \(\mathrm{K}\) or \(\mathrm{^\circ C}\) since a change of one kelvin equals a change of one degree Celsius.

Specific heat capacity. Often we want a property intrinsic to the material rather than the whole object. The specific heat capacity \(c\) (sometimes called specific heat) is the heat required to raise the temperature of one kilogram of the material by one kelvin. It is related to the heat and the mass by

\[
c \;=\; \frac{Q}{m\,\Delta T}
\]

Here \(m\) is the mass in kilograms, and \(c\) has units \(\mathrm{J\,kg^{-1}\,K^{-1}}\). If you know \(c\) and the mass \(m\), the heat capacity of the object is \(C = m\,c\).

Molar heat capacity. If instead you want the heat per mole, use the molar heat capacity \(C_{\mathrm{m}}\). If \(n\) is the number of moles,

\[
C_{\mathrm{m}} \;=\; \frac{Q}{n\,\Delta T}
\]

If you have specific heat \(c\) and the molar mass \(M\) in \(\mathrm{kg\,mol^{-1}}\), then \(C_{\mathrm{m}} = c\,M\) when \(c\) is per kilogram.

Heat supplied experimentally. In many experiments the heat supplied \(Q\) is measured electrically from a heater. If a heater runs at a constant voltage \(V\) and current \(I\) for a time \(t\), the electrical energy delivered is

\[
Q \;=\; V\,I\,t
\]

In more general terms, \(Q = \int P\,\mathrm{d}t\) where \(P\) is power. Be sure units are consistent so that \(Q\) is in joules.

Stepwise procedure to calculate heat capacity or specific heat from an experiment. Step 1. Measure the mass \(m\) of the sample. Step 2. Measure the initial temperature \(T_{\mathrm{i}}\) and the final temperature \(T_{\mathrm{f}}\). Compute the temperature change \(\Delta T = T_{\mathrm{f}} – T_{\mathrm{i}}\). Step 3. Determine the heat added \(Q\). If using an electrical heater, compute \(Q = V I t\). Step 4. Compute the heat capacity of the object using \(C = Q/\Delta T\). Step 5. If you need the specific heat capacity, compute \(c = Q/(m\,\Delta T)\). Step 6. Include corrections for heat losses to the environment if necessary. Typical corrections are measured by a separate empty-calorimeter run or by estimating heat lost during the heating interval.

Worked numerical example. Suppose you have a \(0.500\ \mathrm{kg}\) sample. The initial temperature is \(20.0\ ^\circ\mathrm{C}\) and the final temperature is \(35.0\ ^\circ\mathrm{C}\). A heater supplies \(V = 12.0\ \mathrm{V}\) at \(I = 2.00\ \mathrm{A}\) for \(t = 600\ \mathrm{s}\). Compute the specific heat \(c\).

Compute heat added:

\[
Q \;=\; V\,I\,t \;=\; 12.0 \times 2.00 \times 600 \;=\; 14400\ \mathrm{J}
\]

Compute temperature change:

\[
\Delta T \;=\; 35.0 – 20.0 \;=\; 15.0\ \mathrm{K}
\]

Compute specific heat:

\[
c \;=\; \frac{Q}{m\,\Delta T} \;=\; \frac{14400}{0.500 \times 15.0} \;=\; \frac{14400}{7.50} \;=\; 1920\ \mathrm{J\,kg^{-1}\,K^{-1}}
\]

Compute heat capacity of the object if desired:

\[
C \;=\; m\,c \;=\; 0.500 \times 1920 \;=\; 960\ \mathrm{J\,K^{-1}}
\]

Notes and common pitfalls. Be careful about heat losses to the surroundings. If heat is lost during the experiment, the measured \(Q\) will overestimate the heat absorbed by the sample and lead to an overestimate of \(c\). Use insulating calorimeters, correct for the heat capacity of the container, and perform control experiments when possible. For gases, be explicit whether you want heat capacity at constant volume \(C_{V}\) or constant pressure \(C_{P}\). For an ideal gas, the molar relation is \(C_{P,\mathrm{m}} – C_{V,\mathrm{m}} = R\), where \(R\) is the gas constant. Always report units and the conditions (constant volume or constant pressure) used for the measurement.