KAHIBARO
Table of Contents
Measuring Heat by Mixing and Energy Exchange
Calorimetry is the study of how much heat is transferred during a physical process. In beginner physics, calorimetry usually means using measured masses and temperature changes to determine an unknown quantity, such as a specific heat capacity or a latent heat. The central idea is simple, heat lost by a hotter object is transferred to a colder object, if the system is well insulated.
A calorimeter is the device used for this purpose. In its simplest form, it is just an insulated container that reduces heat exchange with the outside environment. Inside it, substances can be mixed, and the resulting temperature change is measured.
In an ideal calorimetry experiment, the total heat transferred within the system adds to zero:
$$\sum Q = 0$$
This means that heat lost by hot parts equals heat gained by cold parts.
Heat Balance in Calorimetry
The basic heat formula used in many calorimetry problems is
$$Q = mc\Delta T$$
where $Q$ is the heat transferred, $m$ is the mass, $c$ is the specific heat capacity, and $\Delta T = T_f - T_i$ is the temperature change.
If a hot object is placed in cooler water, then the hot object cools and the water warms. Their heat transfers have opposite signs. For two substances in an ideal insulated calorimeter,
$$Q_1 + Q_2 = 0$$
so
$$m_1 c_1 (T_f - T_{i1}) + m_2 c_2 (T_f - T_{i2}) = 0$$
The final temperature $T_f$ is the same for both substances after thermal equilibrium is reached.
Always keep the sign of $\Delta T$ consistent:
$$\Delta T = T_f - T_i$$
If an object cools, then $\Delta T < 0$ and its $Q$ is negative.
If an object warms, then $\Delta T > 0$ and its $Q$ is positive.
Mixing Problems
A common calorimetry situation is mixing two quantities of the same substance, often water at different temperatures. Since the specific heat capacity is the same for both portions, it cancels out.
If mass $m_1$ at temperature $T_1$ is mixed with mass $m_2$ at temperature $T_2$, then
$$m_1(T_f - T_1) + m_2(T_f - T_2) = 0$$
Solving for the final temperature gives
$$T_f = \frac{m_1T_1 + m_2T_2}{m_1 + m_2}$$
This result shows that the final temperature is a mass weighted average of the initial temperatures.
Including the Calorimeter Itself
Real calorimeters also absorb some heat. The container, thermometer, and stirrer may all gain or lose heat. In that case, the heat balance must include them.
If the calorimeter has heat capacity $C_{\text{cal}}$, then its heat transfer is
$$Q_{\text{cal}} = C_{\text{cal}}(T_f - T_{i,\text{cal}})$$
The full energy balance becomes
$$Q_{\text{hot}} + Q_{\text{cold}} + Q_{\text{cal}} = 0$$
Sometimes the calorimeter starts at the same temperature as the cold water, which makes the equation easier.
Do not ignore the calorimeter unless the problem clearly says to neglect it, or states that the calorimeter is ideal.
Determining Specific Heat Capacity
Calorimetry can be used to find the specific heat capacity of an unknown material. A sample of known mass is heated to a known temperature, then placed into water of known mass and initial temperature. After mixing, the final temperature is measured.
If heat absorbed by the calorimeter is neglected, then
$$m_{\text{sample}} c_{\text{sample}} (T_f - T_{\text{sample},i}) + m_w c_w (T_f - T_{w,i}) = 0$$
Solving for the unknown specific heat gives
$$c_{\text{sample}} = -\frac{m_w c_w (T_f - T_{w,i})}{m_{\text{sample}} (T_f - T_{\text{sample},i})}$$
Because the sample usually cools, its temperature change is negative, so the final value of $c_{\text{sample}}$ comes out positive.
Phase Changes in Calorimetry
Calorimetry is also important when melting, freezing, boiling, or condensing occurs. During a phase change, the temperature stays constant while heat is transferred.
In this case, the heat is given by
$$Q = mL$$
where $L$ is the latent heat of the phase change.
For example, if ice at $0^\circ \mathrm{C}$ is added to warm water, several steps may be involved. First the ice melts, then the melted water may warm up to the final temperature. The warm water cools down. Each part must be included in the total heat balance.
A typical expression is
$$Q_{\text{warm water}} + Q_{\text{melt ice}} + Q_{\text{warm melted ice}} = 0$$
That is,
$$m_w c_w (T_f - T_{w,i}) + m_i L_f + m_i c_w (T_f - 0) = 0$$
if the ice starts at $0^\circ \mathrm{C}$.
If the ice begins below $0^\circ \mathrm{C}$, then it must first warm to $0^\circ \mathrm{C}$ before melting. That adds another term.
Step by Step Method
Most calorimetry problems become manageable if handled in a fixed order. First identify all substances involved. Then write the initial and final temperature for each one. Next decide whether each substance warms, cools, or changes phase. After that, write one heat expression for each process and add them together so that the total is zero.
The most important part is choosing the correct expression for each stage. Use $Q = mc\Delta T$ for temperature changes and $Q = mL$ for phase changes.
Common Forms of Calorimetry Equations
| Situation | Heat expression |
|---|---|
| Temperature change | $Q = mc\Delta T$ |
| Calorimeter with heat capacity | $Q = C_{\text{cal}}\Delta T$ |
| Melting or freezing | $Q = mL_f$ |
| Boiling or condensation | $Q = mL_v$ |
| Ideal insulated system | $\sum Q = 0$ |
Example Structure
Suppose a hot metal block is dropped into cooler water. The metal loses heat and the water gains heat. If the calorimeter is neglected, then
$$m_m c_m (T_f - T_{m,i}) + m_w c_w (T_f - T_{w,i}) = 0$$
From this one equation, the unknown could be the final temperature, the mass, or the specific heat, depending on the data given.
If instead ice is added to warm water, then the solution may require checking whether all the ice melts. This is done by comparing the heat available from the warm water with the heat needed to melt the ice.
Experimental Considerations
Real calorimetry experiments are never perfectly isolated. Some heat may escape to the surrounding air. Temperatures may not be measured exactly. A hot object may cool slightly before it is placed into the water. These effects produce error.
To reduce error, the calorimeter should be well insulated, the contents should be mixed gently so the temperature becomes uniform, and the final temperature should be read quickly and carefully.
Visualizing a Simple Calorimeter
Key Ideas to Remember
Calorimetry is based on conservation of energy in thermal processes. The heat lost by one part of a system is gained by another part, provided little energy escapes to the surroundings. The main equations are $Q = mc\Delta T$, $Q = mL$, and the balance condition $\sum Q = 0$. Careful attention to signs, phase changes, and the heat capacity of the calorimeter is essential.