# Temperature, Thermal Equilibrium, and Specific Heat Capacity

## 1. Temperature, Energy, and Heating

Objects contain a type of energy (called internal energy) that determines how hot or cold they are. It is possible to measure how hot an object is by letting some of the energy from that object flow into a thermometer. This flow of energy is called heating. Heating produces a change of temperature; temperature is what is measured by a thermometer.

Figure 1 shows two cubes where the intensity of the red shading represents the amount of internal energy at different points on one of the cubes’ faces. The temperature measured by a thermometer indicates how much energy flows from the point on an object that the thermometer touches. This is not the same thing as measuring how much energy is in the entire object.

The internal energy of an object is distributed throughout that object. The larger an object is, the further this energy will be spread out throughout that object. The further that energy is spread out through an object, the less energy will be available to heat a thermometer placed on any one part of that object.

Given some fixed amount of internal energy, the larger that an object is, the lower its temperature will be, as each part of the object will contain less energy. Speaking of the size of an object is not really accurate. To be correct we must say that the greater the mass of an object, the less temperature the object will have for a given amount of internal energy. Matter is what actually contains internal energy and the mass of an object is the amount of matter in that object.

### Example 1: Temperature, Energy, and Mass Relationship

The temperature of an object is not simply equal to that object’s internal energy divided by that object’s mass. It is true however that for different objects, the temperature of the objects will depend on their ratios of internal energy to mass. The temperature of objects is proportional to their ratios of internal energy to their mass.

1. A brick of mass 1.5 kg contains 3.5 MJ of internal energy. A book of mass 0.85 kg contains 3.5 MJ of internal energy. Which object has the higher temperature?
2. A brick of mass 1.25 kg contains 3.2 MJ of internal energy. A plank of wood of mass 1.25 kg contains 2.6 MJ of internal energy. Which object has the higher temperature?
3. A brick of mass 2.2 kg contains 5.0 MJ of internal energy. A plank of wood of mass 0.50 kg contains 1.5 MJ of internal energy. Which object has the higher temperature?

### Solution

1. Both objects contain the same amount of internal energy. The book has less mass then the brick, meaning that the book’s internal energy is more concentrated and will produce a higher temperature.
2. Both objects have the same mass. The brick contains more internal energy divided over the same mass as the plank of wood. The brick has a higher temperature.
3. The internal energy of the brick divided by its mass has a smaller value than the the internal energy of the bag divided by its mass. The bag has a higher temperature.

## 2. Heating, Cooling, and Thermal Equilibrium

Objects that can heat and cool each other (because they are touching each other for example) are said to be in thermal contact with each other.

Heating and cooling are terms that refer to the flow of internal energy between objects in thermal contact with each other.

• An object is heated when internal energy flows to it, which increases that object’s temperature.
• An object is cooled when internal energy flows from it, which decreases that object’s temperature.

Energy can both flow into and out of an object at the same time. If the amount of energy flowing into an object is equal to the amount of energy flowing out of that object, then the heating and the cooling of the object are equal to each other. When the heating and cooling of an object are equal to each other, then an object’s temperature does not change. An object with an unchanging temperature is said to be in thermal equilibrium.

### Thermal Equilibrium

An object is in thermal equilibrium if it is heated and cooled at the same rate, and so maintains a constant temperature.

The heating of an object can increase and the object can continue to be in thermal equilibrium. If an object increases how much it is cooling by an amount equal to the amount by which its heating increases, then its temperature does not change. We can illustrate this concept by considering a flow of water rather than a flow of energy, as shown in Figure 2.

Figure 2 shows water flow into and out of a bucket that has holes in its side. The faster water flows into the bucket, the higher the water level. The higher the water level, the more holes water can flow through, so the faster the water flows out. If water starts to flow into the bucket faster, then it also flows out of the bucket faster. The water level stays constant as the water flows out as quickly as it flows in.

In Figure 3, we see an object that is in thermal equilibrium, being heated and cooled at the same rate. The heating rate of the object then increases. The increase in heating rate increases the temperature of the object. The increase in the temperature of the object increases the rate at which it heats its surroundings, as the surroundings have not increased in temperature, as shown in Figure 4.

The object continues to be in thermal equilibrium at its higher temperature.

Objects in thermal contact with each other and which have the same temperature as each other are in thermal equilibrium with each other, as shown in Figure 5.

### Example 2: Maintaining Thermal Equilibrium

A lake with a steady temperature of is being heated by sunlight. The lake is also cooling itself by heating the air in contact with it. The heating of the lake and the air occur at a rate of 545 MW. The lake is not losing internal energy in any other way. Later in the day, the sunlight increases its heating power to 870 MW and the lake’s temperature rises to a steady . At what rate is the lake then heating the air in contact with it?

### Solution

At a steady temperature of , the lake is being heated by 545 MW and cooling itself by 545 MW. The heating and cooling are equal to each other, so the lake’s net flow of internal energy is zero. The lake maintains a constant temperature as the magnitude of its internal energy does not change.

The heating by sunlight then increases to 870 MW. The lake now receives more energy by heating than it loses by cooling, causing its temperature to increase to . At this higher temperature, the lake cools itself faster than it did at its lower temperature as the lake heats the air around it at a higher rate. If the lake temperature of remains steady then the magnitude of the energy that the lake loses by cooling must be equal to the magnitude of the energy that the lake gains by heating in the same amount of time. As the sunlight is heating the lake with 870 MW, the lake must be cooling itself by 870 MW.

## 3. Thermal insulation

How quickly an object is heated or cooled is affected by how easily internal energy flows through the material that an object is made of. All materials have their own value of thermal conductivity. The higher this value, the more easily internal energy can flow through them. Materials with low thermal conductivity values are good thermal insulators.

Again, this can be understood by comparison with flow of water, as Figure 6 shows.

If an object is well insulated, then it is possible to heat that object more quickly than the object will be cooled. If an object is perfectly insulated then it cannot be cooled at all. When this is the case, then all the energy heating that object remains in the object.

### Example 3: Effect of Insulation

An insulated object is heated at a rate of 250 J for 140 seconds. The object is well insulated and 240 seconds after heating stops the object still retains J of the energy that heated it.

How much heat escaped the object as it cooled after heating stopped?

### Solution

Heating the object added 250 J J.

J of heat is retained. The heat lost during cooling J J. J of heat escaped the object.

## 4. Specific Heat Capacity

The specific heat capacity of a substance determines how much energy supplied by heating will increase the temperature of an object made of 1 kilogram of that material by . This definition assumes that all the energy heating an object remains in the object. Figure 7 illustrates this definition.

Heating adds energy to an object. The temperature increase resulting from this heating equals the energy added divided by the heat capacity of the object. An object’s heat capacity is equal to that objects mass in kg multiplied by the specific heat capacity of the substance from which the object is made.

### Specific Heat Capacity

The change of an object’s internal energy equals the object’s mass multiplied by its specific heat capacity, multiplied by the change in the object’s temperature. In equation form we have where, is the change of a heated or cooled object’s internal energy, is the mass of a heated or cooled object, is the temperature change of a heated or cooled object, and is the specific heat capacity of a heated or cooled object.

The unit of specific heat capacity is .

### Example 4: Calculating Internal Energy Change

Internal energy cannot be measured directly, but can be calculated through temperature measurements by using the specific heat capacity equation.

A perfectly insulated block of iron with a mass of 3.28 kg is heated. Iron has a specific heat capacity of 510 . If the iron’s initial temperature is and this rises to , how much internal energy is added to the block of iron?

### Solution

Using the equation