## Heat capacity and specific heat

The **heat capacity** of a body is the amount of heat energy necessary to raise the temperature of an object by one degree. Imagine blocks of the same mass made of different metals (see Figure

**Figure 1**

Hot blocks of different metals, at the same temperature, melt different amounts of ice.

The heat capacity *(C)* per unit mass *(m)* is called **specific heat** ( *c*):

*Q* units of heat are added to m kg of a substance, changing the temperature by Δ *T*. The specific heats have been determined for many materials and can be found in tables.

## Mechanical equivalent of heat

The **calorie** is defined as the amount of energy required to raise 1 gram of water 1 degree. (This energy is slightly dependent upon the temperature of the water, so the temperature change is usually defined from 14.5 degrees to 15.5 degrees Celsius.) The **kilocalorie** is the amount of heat energy needed to raise 1 kilogram of water by 1 degree Celsius. (Food calories are kilocalories.) In SI units, the calorie equals 4.184 joules. The U.S. engineering unit of heat is the **British thermal unit** (BTU). It is related to the calorie and the joule: BTU = 252 calories = 1.054 kJ. These reversible conversions of heat energy and work are called the **mechanical equivalent of heat**.

## Heat transfer

The heat energy *(Q)* transferred into or out of a system is given by *Q* = *mc*Δ *T*. The temperature change is positive for a gain in heat energy and negative for heat removed from the object. When applying this expression in heat exchange problems, assume that the objects in thermal contact are isolated from their surroundings—completely insulated.

## Calorimetry

If a substance in a closed container loses heat, then something else in the container gains an equal amount of heat. A **calorimeter** is a device that utilizes the transfer of heat to determine the specific heat of a substance. A known mass of a substance whose specific heat is unknown is heated to a certain temperature and then placed in a container containing a liquid (usually water) of known mass, specific heat, and temperature. After thermal equilibrium is reached, the specific heat of the unknown can be determined.

**Example 1:** Consider a block of hot metal, with mass *m* _{m }and original temperature *T* _{mo }, which is dropped into a mass of cool water of mass *m* _{w }with beginning temperature *T* _{wo }. If the final temperature is *T*, what is the specific heat of the metal?

**Solution**: All of the heat lost by the metal is gained by the water because the system is isolated. The heat lost by the unknown is *Q* _{m }= *m* _{m }*c* _{m }Δ *T* _{m }= *m* _{m }*c* _{m }( *T* _{mo }− *T* _{f }), and the heat gained by the water is *Q* _{w }= *m* _{w }*c* _{w }Δ *T* _{w }= *m* _{w }*c* _{w }( *T* _{f }− *T* _{wo }). The temperature differences have been written so that they are both positive quantities. The final temperature of the water will be greater than its original temperature because it is warming. The final temperature of the metal will be less than its original temperature. The objects attain thermal equilibrium, and so the final temperatures are the same. (The specific heat for water has the value of 1 cal/g.k.)

## Latent heat

A **change of phase** occurs when an object changes from one physical state to another. The common **physical states** are solid, liquid, or gas. Some examples of phase changes are from a liquid to a solid (freezing) or from a liquid to a gas (boiling).

The plot shown in Figure 2 **latent heat** *(L)*: *Q* = *mL*.

**Figure 2**

Phase changes of water as heat is added.

## The heat of fusion

The value of latent heat *(L)* depends upon the particular phase change as well as the properties of the substance. The **heat of fusion** is the heat required for a phase change from a solid to a liquid. If the substance is originally in liquid form, the heat of fusion is the heat released when the substance changes from a liquid to a solid. The latent heat of fusion for water at atmospheric pressure is 3.34 × 10 ^{5} J/kg. The heat required to melt 1 gram of water at 0°C is

## The heat of vaporization

The latent **heat of vaporization** concerns the phase change between the liquid and gaseous states. The heat of vaporization for water is 2.26 × 10 ^{6} J/kg. The amount of heat necessary to convert 1 gram of water to steam at 100°C is *Q* = *mL* _{v }= (10 ^{−3}kg)(2.26 × 10 ^{6} J/kg) = 2.26 × 10 ^{3} J. Continued addition of heat to steam will cause the steam to be superheated, to attain a higher temperature than 100°C.

Note from the graph in Figure

## Methods of heat transfer: Conduction, convection, and radiation

Heat energy can be transferred from one location to another by one of three methods: conduction, convection, and radiation.

The metal handle of an iron skillet placed on a heated burner gets hot by conduction. **Conduction** occurs when the heat travels through the heated solid. The **transfer rate** *(H)* is the ratio of the amount of heat per time transferred from one location in an object to another *H* = *Q*/Δ *t*, where *H* has units of watts or J/s, when *Q* is in joules, and Δ *t* is in seconds. The temperature between two parts of the conducting medium—the pan bottom and the handle—must be different for conduction to take place. The formula for heat conduction from one side to another of a slab with thickness *L* and cross‐sectional area *A* is given by

*T* _{2} to *T* _{1} and *T* _{2} > *T* _{1}, as shown in Figure 3

**Figure 3**

Heat flows from regions of higher temperature to lower.

The constant *(k)*, called **thermal conductivity**, is found in tables listing properties of materials. The fact that different materials have different *k* values explains why the metal shelf of a refrigerator feels colder than the food on it even if both are at thermal equilibrium. The conductivity constant is relatively large for metals, and the metal feels colder because the heat is conducted away from the hand more quickly by metal than by other materials.

Heat transported by the movement of a heated substance is a result of **convection**. The most common example of convection is the warmed mass of air rising from a heater or fire.

The third mechanism for heat transfer is **radiation** in the form of electromagnetic waves. Radiant energy from the sun warms the earth. The rate at which an object emits radiant energy is proportional to the fourth power of its absolute temperature. The **Stefan‐Boltzmann law**, which describes the relationship, is written *P* = σ *AeT* ^{4}, where *P* is the power radiated in watts, σ is a constant equal to 5.6696 × 10 ^{−s} W/m ^{2}K ^{4}, *A* is the surface area of the object in *m* ^{2}, *T* is the absolute temperature, and *e* is the **emissivity constant**, which varies from 0 to 1 depending upon the properties of the surface.

The thermos bottle, or Dewar flask, is an object that minimizes heat transfer by conduction, convection, and radiation. The flask is constructed of double‐walled Pyrex glass with silvered inner walls. The space between the walls is evacuated to reduce heat transfer by conduction and convection. The silvered walls reflect most of the radiant heat to cut heat transfer by radiation. The container is effectively used to store either cold or hot liquids for long periods of time.