Calorimetry

Heat Capacity

When heat is transferred to an object, the temperature of the object increases. Heat transfer is a form of energy transfer, thus heat transfer has units of joules (J). When heat is removed from an object, the temperature of the object decreases. The relationship between the heat transferred ( q ) and the change in temperature ( ΔT ) is

q = C ΔT   =   C ( T_f   -   T_i )

The proportionality constant C in this equation is called the heat capacity. The heat capacity is the amount of heat required to raise the temperature of an object or substance one degree. The temperature change is the difference between the final temperature ( T_f ) and the initial temperature ( T_i ).

Note that the heat capacity always has a positive value, but ΔT can be positive or negative. The system can become hotter (positive ΔT ) or colder (negative ΔT ) depending upon whether heat is transferred into or out of the system. Consequently, q can also be either positive or negative. In chemistry, a positive value for q indicates heat is transferred into the object. A negative value means heat is transferred out of the object.

The heat capacity is an extensive property. That is, the heat capacity depends upon the amount of substance present. Consider a sample of water. It takes more heat to increase the temperature of 1 kg of water 1 °C than to increase the temperature of 1 g of water 1 °C. It is impractical to list separate heat capacities for every quantity of water. When dealing with variable amounts of material, an intensive measure of the heat capacity is preferred. Unlike extensive properties, intensive properties do not depend upon the amount of material. One common intensive version of the heat capacity is the specific heat capacity ( s ), which is the heat capacity of one gram of a substance.

For a sample of water ( subscript w ), the specific heat capacity s_w is related to the heat capacity C_w by

The symbol m_w represents the mass of water.

Near room temperature, the specific heat capacity of water is s_w = 4.184 J °C ^-1 g ^-1

The table below summarizes the thermodynamic properties related to heat transfer and heat capacity.

Calorimetry

A calorimeter is an experimental device in which a chemical reaction or physical process takes place. The calorimeter is well-insulated so that, ideally, no heat enters or leaves the calorimeter from the surroundings. For this reason, any heat liberated by the reaction or process being studied must be picked up by the calorimeter and other substances in the calorimeter.

A thermometer is typically inserted in the calorimeter to measure the change in temperature that results from the reaction or physical process. A stirrer is employed to keep the contents of the calorimeter well-mixed and to ensure uniform heating.

Like any other physical object, the calorimeter itself has a heat capacity. The calorimeter must absorb heat to increase its temperature or lose heat to lower its temperature. As described above, the heat capacity of the calorimeter, C_cal , relates the temperature change for the calorimeter to the heat transferred to the calorimeter, q_cal . In these symbols, the subscript cal indicates that the property ( q for heat or C for heat capacity ) applies to the calorimeter (as opposed to some other object or process).

q_cal   =   C_cal ΔT   =   C_cal ( T_f   -   T_i )

Because the calorimeter comes as a fixed unit, its heat capacity has a fixed value.

If a calorimeter is perfectly insulating, then there is no heat transfer between the calorimeter and the surroundings. Any heat released by something inside the calorimeter must be absorbed by something else inside the calorimeter or by the calorimeter itself.

In the experiments below, a hot metal ball is dropped into water in a calorimeter and the system is allowed to reach a uniform final temperature. There are three objects in this experiment: the metal ball, the water and the calorimeter. Each object has a heat flow: q_m for the metal ball, q_w for the water and q_cal for the calorimeter. q_m will be negative, because the hot metal ball is cooled by the water. q_w and q_cal are each positive, because both the water and calorimeter are warmed by heat from the hot metal ball. If the calorimeter is perfectly insulated from the surroundings, these three heat flows must add to zero.

0   =   q_m   +   q_w   +   q_cal

Each of these heat flows can be mathematically related to the corresponding heat capacity and the temperature change for that object, as described above.

Experiment

The goal of this experiment is to determine the specific heat capacity of copper metal. As described above, a hot ball of copper metal is dropped into water in a calorimeter.

The initial temperature of the copper is 100.0 °C.

Both the water and the calorimeter are near room temperature initially ( T_i ). After the copper metal has thermally equilibrated with the water and calorimeter, all three objects (copper, water and calorimeter) have the same final temperature ( T_f ). The sum of the heat flow for each object must add to zero, because no heat can enter or leave the calorimeter.

0   =   q_m   +   q_w   +   q_cal

For each object, its heat flow can be related to a heat capacity and the temperature change. For the water and the copper, it is convenient to determine the heat capacity for the substance's specific heat capacity and the mass of the object.

q_Cu   =   C_Cu   ( T_f   -   100.0 °C )   =   m_Cu   s_Cu   ( T_f   -   100.0 °C )

q_w   =   C_w   ( T_f   -   T_i )   =   m_w   s_w   ( T_f   -   T_i )

q_cal   =   C_cal   ( T_f   -   T_i )

A thermometer is placed in the water to measure the temperature of the water. At thermal equilibrium, the water and the calorimeter have the same temperature. Measurements are only made when the water is in thermal equilibrium with the calorimeter.

The mass of copper and mass of water are known by how the experiment is set up. The initial temperature T_i of the water and calorimeter is measured before the copper ball is dropped into the water. After the copper ball has been dropped into the water and the system has equilibrated, the final temperature T_f is measured.

While the specific heat capacity of water is known, the heat capacity of the calorimeter is not known and must be measured experimentally. In Part 1 of the experiment, the value of C_cal will be determined by performing the experiment using iron instead of copper. The specific heat capacity of iron is s_Fe = 0.450 J °C ^-1 g ^-1. Data from the measurements with iron permit experimental determination of C_cal. Once the heat capacity of the calorimeter has been determined, measurements involving copper metal (Part 2) allow the specific heat capacity of copper to be determined.

Part 1: Determination of the Heat Capacity of the Calorimeter

Perform the experiment using a hot iron ball. Use the changes in temperature to calculate q_Fe and q_w. Then calculate q_cal using

0   =   q_Fe   +   q_w   +   q_cal

Use q_cal and ΔT for the calorimeter to determine C_cal.

Specify the amount of iron (between 10 and 100 g) and the amount of water (between 30 and 80 g). Use the thermometer to read the initial temperature, T_i , of the water and calorimeter. Run the simulation. After the iron ball has been dropped into the water and thermal equilibrium has been reached, read the final temperature, T_f .

Perform the experiment several times using different masses of iron and water. You should always get the same value for C_cal, within experimental error.

Part 2: Determination of the Specific Heat Capacity of Copper

Once the heat capacity of the calorimeter has been determined, perform the experiment using copper metal. Use the changes in temperature to calculate q_cal and q_w. Use your average value for C_cal from Part 1 to determine q_cal . Then calculate q_Cu using

0   =   q_Cu   +   q_w   +   q_cal

Use q_Cu , m_Cu and ΔT for copper to determine s_Cu.

Specify the amount of copper (between 10 and 100 g) and the amount of water (between 30 and 80 g). Use the thermometer to read the initial temperature, T_i , of the water and calorimeter. Run the simulation. After the copper ball has been dropped into the water and thermal equilibrium has been reached, read the final temperature, T_f .

Perform the experiment several times using different masses of iron and water. You should always get the same value for C_cal, within experimental error. When you have a value for the specific heat capacity of copper, check your answer with the accepted values.