# Thermal energy and specific heat

#### Prerequisites

Given a thermometer — a way of measuring how hot or cold something is -- we can begin a series of experiments to observe how objects share their thermal energies when they are put together. Here's an example:

- Mix two equal amounts of water at different temperatures.
- Mix two different amounts of water at different temperatures.
- Mix two different kinds of substances at different temperatures

The result are three important observations:

- For the
**same substance, equal amounts**, the temperature of the mixture is the average of the two initial temperatures:

$$T_{final} = {1 \over 2}(T_1 + T_2)$$ - For the
**same substance, different amounts**, the temperature of the mixture is the weighted average of the two initial temperatures, weighted by the fraction of the total mass that has each initial temperature:

$$T_{final} = \bigg(\frac{m_1}{m_1 + m_2}\bigg)T_1 + \bigg(\frac{m_2}{m_1 + m_2}\bigg)T_2$$ - For different substances, the above rule does NOT hold. Some substances seem to have more "thermal inertia" than others.

In order to figure out what to do in the general case, let's build a mechanistic model of what's going on when two objects in good contact come to the same temperature.

## An energy model of shared temperature

To see what's going on when two objects in good contact come to the same temperature, let's create a model based on our understanding of what happens. Here are the basic ideas of the model:

- Temperature is a measure of the average kinetic energy of the molecules of a substance.
- When two objects come into contact, collisions at the boundary between the faster moving molecules of the hotter substance and the slower moving molecules of the cooler substance will tend to make the slower ones move faster and the faster ones move slower -- so to warm up the cooler substance and cool down the hotter substance.
- The total thermal energy that leaves the hotter substance will be equal to the total energy that enters the cooler substance, since energy is conserved.
- How that energy is converted to temperature will depend on how big the object is (how much mass, since we're talking about kinetic energy) and what kind of material it's made of.

The thermal energy transferred from one object to another by contact is referred to as ** heat** and is usually labelled with the symbol

*Q*. If we conjecture that the amount of thermal energy in an object is proportional to the temperature times the mass (not quite true, but OK if we are looking at small changes), then we would conjecture as a result of these experiments:

$$\Delta T = \frac{Q}{mc}$$

The temperature change produces by an amount of heat energy, *Q*, entering a mass *m* is a result of sharing that heat energy over each part of the mass, and must be corrected by a constant, *c*, that depends on what the substance is. The constant *c* is called the *specific heat* of the particular material. Though this is the way of looking at the equation that says "heat energy transfer causes temperature change", the equation is usually written in this way:

$$Q = mc \Delta T$$

Note that what we are doing is similar to what we are doing in a number of situations when we try to extract the properties that depend on external circumstances and specific objects from the property of matter that depends on just what kind of substance it is. We did this when we defined density, Young's modulus, and bulk modulus, for example.

## Applying the equation in specific examples

This now makes it easy to figure out what happens when we put two substances together. Suppose we have the same substance of masses *m*_{1} and *m*_{2} at temperatures $T_1$ and $T_2$ respectively and they come to a final temperature $T$. For concreteness, let's assume that 1 is hotter and 2 is colder and that both are water. Then the heat entering the cold water (2) is a positive quantity and the heat entering the hot water (1) is a negative quantity (that is, it leaves, doesn't enter). The change in the temperature of the cold water is positive and that of the hot water is negative. The equation for heat flow tells us

$$Q_{1 \rightarrow 2} = -Q_{2 \rightarrow 1}$$

$$m_1c_1(T - T_1) = -m_2c_2(T - T_2)$$

Check why we put the signs the way we did!

Now since the objects are both water, the *c*'s are the same and cancel out. We can now solve for *T*:

$$m_1(T - T_1) = -m_2(T - T_2)$$

$$m_1T - m_1T_1 = -m_2T + m_2T_2$$

$$m_1T +m_2T = m_1T_1 + m_2T_2$$

$$(m_1 +m_2)T = m_1T_1 + m_2T_2$$

$$T = \bigg(\frac{m_1}{m_1 +m_2}\bigg)T_1 + \bigg(\frac{m_2}{m_1 +m_2}\bigg)T_2$$

Exactly the result we found experimentally! The final temperatures are the original temperatures, weighted by the fraction of the total mass they each represent.

If we include the specific heats as possibly different, the result would turn out to be

$$T = \bigg(\frac{m_1c_1}{m_1c_1 +m_2c_2}\bigg)T_1 + \bigg(\frac{m_2c_2}{m_1c_1 +m_2c_2}\bigg)T_2$$

The final temperature is the original temperatures weighted by the fraction of the total "*mc*" they represent. This combination -- mass times specific heat -- is called the ** heat capacity **of the object.

Though it might not seem so from the messy algebra above, it is a lot easier to keep signs straight in specific examples if you do NOT try to memorize the above equation but rather use the conceptual equation about how energy flow is related to temperature change ($Q = mcΔT$) and work it out.

### A note on units

When this stuff was first studied in the early 1800's, the relation with molecular motions and mechanical energy was not understood. As a result, the "whatever it was" that was transferred from one object to another when the temperature changed was given its own definition and defined by its own standard.

The amount of "heat" that needed to be transferred to change the temperature of 1 gram water by 1°C was taken to be **1 calorie**. (Actually, this was specified to be when the water was at 4°C since the specific heat of water changes a little with temperature. We will ignore this.) This makes the amount of "heat" needed to be transferred to change the temperature of 1 kg of water by 1°C equal to 1000 calories. This is sometimes called a **kilocalorie** or just a "big" Calorie (a calorie with a capital C). This is very confusing. It is much better to go to the standard unit of Joule for energy. The conversion is

1 cal = 4.186 Joules

1 kcal = 4186 Joule

Although this establishes the basic concept of temperature and its relation with energy, the situation is more complex. It often takes thermal energy to change the state of a substance — the arrangement and therefore the potential energy associated with its molecular interactions. Thus, it takes thermal energy to melt ice or turn water to steam — without changing the temperature. Further, our understanding of the meaning of temperature develops as we understand how it relates to molecular motion. These and other issues are discussed in the follow ons.

Joe Redish 11/19/11, 11/30/12

#### Follow-ons

Last Modified: March 7, 2019