## Temperature and thermal expansion

6-22-98

We'll shift gears in the course now, moving on to thermal physics.

### Temperature scales

In the USA, the Fahrenheit temperature scale is used. Most of the rest of the world uses Celsius, and in science it is often most convenient to use the Kelvin scale.

The Celsius scale is based on the temperatures at which water freezes and boils. 0°C is the freezing point of water, and 100° C is the boiling point. Room temperature is about 20° C, a hot summer day might be 40° C, and a cold winter day would be around -20° C.

To convert between Fahrenheit and Celsius, use these equations:

The two scales agree when the temperature is -40°. A change by 1.0° C is a change by 1.8° F.

The Kelvin scale has the same increments as the Celsius scale (100 degrees between the freezing and boiling points of water), but the zero is in a different place. The two scales are simply offset by 273.15 degrees. The zero of the Kelvin scale is absolute zero, which is the lowest possible temperature that a substance can be cooled to. Several physics formulas involving temperature only make sense when an absolute temperature (a temperature measured in Kelvins) is used, so the fact that the Kelvin scale is an absolute scale makes it very convenient to apply to scientific work.

### Measuring temperature

A device used to measure temperature is called a thermometer, and all thermometers exploit the fact that properties of a material depend on temperature. The pressure in a sealed bulb depends on temperature; the volume occupied by a liquid depends on temperature; the voltage generated across a junction of two different metals depends on temperature, and all these effects can be used in thermometers.

### Linear thermal expansion

The length of an object is one of the more obvious things that depends on temperature. When something is heated or cooled, its length changes by an amount proportional to the original length and the change in temperature:

The coefficient of linear expansion depends only on the material an object is made from.

If an object is heated or cooled and it is not free to expand or contract (it's tied down at both ends, in other words), the thermal stresses can be large enough to damage the object, or to damage whatever the object is constrained by. This is why bridges have expansion joints in them (check this out where the BU bridge meets Comm. Ave.). Even sidewalks are built accounting for thermal expansion.

Holes expand and contract the same way as the material around them.

### Example

Consider a 2 m long brass rod and a 1 m long aluminum rod. When the temperature is 22 °C, there is a gap of 1.0 x 10-3 m separating their ends. No expansion is possible at the other end of either rod. At what temperature will the two bars touch?

The change in temperature is the same for both, the original length is known for both, and the coefficients of linear expansion can be looked up in a table.

Both rods will expand when heated. They will touch when the sum of the two length changes equals the initial width of the gap. Therefore:

So, the temperature change is:

If the original temperature was 22 °C, the final temperature is 38.4 °C.

### Thermal expansion : expanding holes

Consider a donut, a flat, two-dimensional donut, just to make things a little easier. The donut has a hole, with radius r, and an outer radius R. It has a width w which is simply w = R - r.

What happens when the donut is heated? It expands, but what happens to the hole? Does it get larger or smaller? If you apply the thermal expansion equation to all three lengths in this problem, do you get consistent results? The three lengths would change as follows:

The final width should also be equal to the difference between the outer and inner radii. This gives:

This is exactly what we got by applying the linear thermal expansion equation to the width of the donut above. So, with something like a donut, an increase in temperature causes the width to increase, the outer radius to increase, and the inner radius to increase, with all dimensions obeying linear thermal expansion. The hole expands just as if it's made as the same material as the hole.

### Volume thermal expansion

When something changes temperature, it shrinks or expands in all three dimensions. In some cases (bridges and sidewalks, for example), it is just a change in one dimension that really matters. In other cases, such as for a mercury or alcohol-filled thermometer, it is the change in volume that is important. With fluid-filled containers, in general, it's how the volume of the fluid changes that's important. Often you can neglect any expansion or contraction of the container itself, because liquids generally have a substantially larger coefficient of thermal expansion than do solids. It's always a good idea to check in a given situation, however, comparing the two coefficients of thermal expansion for the liquid and solid involved.

The equation relating the volume change to a change in temperature has the same form as the linear expansion equation, and is given by:

The volume expansion coefficient is three times larger than the linear expansion coefficient.

### Temperature, internal energy, and heat

The temperature of an object is a measure of the energy per molecule of an object. To raise the temperature, energy must be added; to lower the temperature, energy has to be removed. This thermal energy is internal, in the sense that it is associated with the motion of the atoms and molecules making up the object.

When objects of different temperatures are brought together, the temperatures will tend to equalize. Energy is transferred from hotter objects to cooler objects; this transferred energy is known as heat.

### Specific heat capacity

When objects of different temperature are brought together, and heat is transferred from the higher-temperature objects to the lower-temperature objects, the total internal energy is conserved. Applying conservation of energy means that the total heat transferred from the hotter objects must equal the total heat transferred to the cooler objects. If the temperature of an object changes, the heat (Q) added or removed can be found using the equation:

where m is the mass, and c is the specific heat capacity, a measure of the heat required to change the temperature of a particular mass by a particular temperature. The SI unit for specific heat is J / (kg °C).

This applies to liquids and solids. Generally, the specific heat capacities for solids are a few hundred J / (kg °C), and for liquids they're a few thousand J / (kg °C). For gases, the same equation applies, but there are two different specific heat values. The specific heat capacity of a gas depends on whether the pressure or the volume of the gas is kept constant; there is a specific heat capacity for constant pressure, and a specific heat capacity for constant volume.

### Example

0.300 kg of coffee, at a temperature of 95 °C, is poured into a room-temperature steel mug, of mass 0.125 kg. Assuming no energy is lost to the surroundings, what does the temperature of the mug filled with coffee come to?

Applying conservation of energy, the total change in energy of the system must be zero. So, we can just add up the individual energy changes (the Q's) and set the sum equal to zero. The subscript c refers to the coffee, and m to the mug.

Note that room temperature in Celsius is about 20°. Re-arranging the equation to solve for the final temperature gives:

The temperature of the coffee doesn't drop by much because the specific heat of water (or coffee) is so much larger than that of steel. This is too hot to drink, but if you leave it heat will be transferred to the surroundings and the coffee will cool.

### Changing phase; latent heat

Funny things happen when a substance changes phase. Heat can be transferred in or out without any change in temperature, because of the energy required to change phase. What is happening is that the internal energy of the substance is changing, because the relationship between neighboring atoms and molecules changes. Going from solid to liquid, for example, the solid phase of the material might have a particular crystal structure, and the internal energy depends on the structure. In the liquid phase, there is no crystal structure, so the internal energy is quite different (higher, generally) from what it is in the solid phase.

The change in internal energy associated with a change in phase is known as the latent heat. For a liquid-solid phase change, it's called the latent heat of fusion. For the gas-liquid phase change, it's the latent heat of vaporization, which is generally larger than the latent heat of fusion. Latent heats are relatively large compared to the heat required to change the temperature of a substance by 1° C.

If you use the sum-of-all-the-Q's equals zero equation, you have to be careful with the heat associated with something changing phase because you need to put it in with the appropriate sign. If heat is going into a substance changing phase, such as when it's melting or boiling, the Q is positive; if heat is being removed, such as when it's freezing or condensing, the Q is negative. We don't have to worry about the signs for the heat required to change temperature, because the sign is already built in to the change in temperature.

Note that a change in phase takes place only under the right conditions. Water, for example, doesn't freeze at 10 °C, at least not at atmospheric pressure. If you had water at that temperature, you would first need to cool it to the melting point, 0 °C, before it would start to freeze.

If you're putting in heat from an outside source, the sum-of-all-the-Q's equation becomes: