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# Thermal Expansion, Ideal Gas Law, and Kinetic Theory of Gases for AP Physics B

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By McGraw-Hill Professional
Updated on Feb 11, 2011

Practice problems for these concepts can be found at:

Thermodynamics Practice Problems for AP Physics B

### Thermal Expansion

Things get slightly bigger when they heat up. We call this phenomenon thermal expansion. This is why a thermometer works. The mercury in the thermometer expands when it gets hotter, so the level of mercury rises. When it gets colder, the mercury shrinks and the level of mercury drops. The equation describing thermal expansion follows:

This formula describes what is called linear expansion. It says that the change in an object's length, ΔL, is equal to some constant, α, multiplied by the object's original length, L0, multiplied by the change in the object's temperature.

The constant α is determined experimentally: it is different for every material. Metals tend to have high values for α, meaning that they expand a lot when heated. Such insulators as plastic tend to have low values for α. But on the AP exam, you will never be expected to have memorized the value of α for any particular material. Here's an example involving thermal expansion.

1. You should heat the ring. You might initially think that because you want the inner diameter of the ring to expand, you want the aluminum itself to shrink. However, when a material expands, it expands in all directions—thus, the hole will expand by the same amount as the surrounding metal when heated.
2. In the length expansion equation, ΔL is the amount by which we need the diameter of the ring to increase, 0.1 mm. Remember to convert to meters before solving.
3. 0.0001 m = (2.5 × 10–5 °C–1)(.050 m)(ΔT)

Solving for ΔT, we find that we need to increase the ring's temperature by 80°C. So the final temperature of the ring is (20 + 80) = 100°C.

Two more notes: (1) Thermal expansion equations assume that length changes will be small compared to the size of the object in question. For example, the object above only expanded by a few tenths of a percent of its original size. Objects will not double in size due to thermal expansion. (2) If the standard unit of temperature is the kelvin, why did we use degrees Celsius in the example? It is true that if an equation includes the variable T for temperature, you must plug in temperature in kelvins. The ideal gas law is one such equation. However, the equation for thermal expansion includes a change in temperature, not an absolute temperature. A change of one degree Celsius is equivalent to a change of one kelvin; so if an equation needs only a temperature change, then either unit is acceptable.

### Rate of Heat Transfer

Heat will flow across a material when there is a temperature difference between two positions on that material. Heat will flow until the material has the same temperature everywhere, when the material is said to have reached thermal equilibrium. The bigger the temperature difference, the larger the heat transfer rate. You could probably guess at the other properties of the material that affect the rate of heat transfer:

• the nature of the material (metals carry heat better than non-metals)
• the cross-sectional area of the substance (more area makes it easier for heat to flow)
• the distance across the material between the two positions with a temperature difference (the longer this distance, the slower the heat transfer)

This can be summed up in the equation for the heat transfer rate H:

where k is the thermal conductivity of the substance, A is the cross-sectional area, ΔT is the temperature difference, and L is the length of the material.

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