The answer is yes, but not directly. There is one more step in the process that makes it all work.
Heat energy largely means the random jiggling of the molecules in a substance, and the temperature is a measure of the average energy per molecule. Nearly all substances, particularly gases, will expand (if they can) when they are heated up. You can stop this expansion by putting the gas in a strong container; then the pressure goes up because the same number of molecules are in the container, but they are hitting the sides more often and they are hitting harder than before. If the container has a turbine attached to an opening leading to the outside air at lower pressure and temperature, then the gas will escape and spin the turbine. It is really the pressure difference between one side of the turbine and the other and the fact that the gas flows from the high-pressure region to the low-pressure region that makes the turbine spin. But the reason the pressures were different in the first place was because one side was heated up. As the gas escapes to the low-pressure region, it expands and cools off.
There are lots and lots of variations on this theme. If you boil water, it expands quite a lot from its compact liquid state to the space-taking vapor state steam. In fact, this is how many turbines work in the real world; the container being heated contains water which is boiled to make steam to turn the turbine. A closed-loop system will recondense the water somewhere else that's cold to make water again and pump it back into the container to be boiled again. This illustrates one very important feature of the whole process. Heat energy can only be used to turn turbines if there is a difference in temperature from one place to another, and heat energy is allowed to flow from the hot place to the cold place. Then some fraction of the energy can be used to spin a turbine. If everything is very hot but all at the same temperature, then the heat energy cannot be used to make the turbine spin.
A special case of a heat-driven turbine is the radiometer, which can be found in novelty stores. Here's a picture of one:
The glass bulb has a partial vacuum (very few gas molecules in it, but itís not completely empty). The vanes each have one black side and one shiny side. When you shine light on it, the black sides of the vanes get hotter than the shiny sides because they absorb the light better. Then when gas molecules come wandering by and strike the black side, they get a strong kick because the random jiggling of the atoms in the vane gives some extra energy to gas molecules colliding with them. On the shiny side, the same number of gas molecules wander by and collide with it too, but they donít come off so fast on average because the atoms on the shiny side arenít jiggling so fast. Newtonís third law, says that the force the gas molecule exerts on the vane is just exactly as big as the force the vane exerts on the gas molecule. So the vanes will start spinning in recoil to all the gas molecules kicked away from the hot, black sides. This is one other way that heat energy can move a turbine. It again relies on temperature differences -- the light source needs to be very hot (the sun will do fine, or a light bulb) to emit the light, and the radiometer itself takes advantage of temperature differences inside.
(published on 10/22/2007)