What are the differences between isothermal expansion and adiabatic expansion?

Thermodynamics is the study of heat and work. Heat and work are ways to transfer energy to and from a system. Internal energy -- the energy of molecular motion -- changes as heat and work are added to or taken away from a system. Thermo variables:

U -- Internal Energy (really, internal motion of molecules)
Q -- Heat (in calories)
W -- Work (in Joules) Note: 1000 cal = 4186 joules

U = Q - W
Isothermal and adiabatic systems are special cases of the first law.

ISOTHERMAL -- No change in temperature occurs during a thermodynamic exchange and therefore U = 0. The First Law reduces to Q = W.

In this case, work and heat are equivalent.

For a good example of an isothermal exchange think of some guy with emphysema blowing up a balloon very, very slowly.The expansion is SO SLOW that no change in temperature occurs and the internal energy is static. Almost like watching paint dry.

ADIABATIC -- No change in heat occurs during a thermodynamic exchange and therefore Q = 0. The First Law reduces to U = -W.

In this case, internal energy depends entirely on work. The equations for this case can be tricky. Here's the way to avoid ambiguity (and wrong answers on tests) --

(1) U = - (+W)
In this equation Work is positive and is being done BY the system on the surroundings. Hence internal energy is negative

(2) U = - (-W)
In this equation Work is negative and is being done ON the system by the surroundings. Hence internal energy is positive.

For a good example of an adiabatic exchange think of a balloon attached to a helium pump and QUICKLY blown up. So quickly that the heat is faked out and doesn't have time to react. The heat's like, "Whoa, dude, what just happened?"

Yet a third type of thermodynamic system is ISOVOLUMETRIC where NO change in volume occurs during a thermodynamic exchange and therefore W = 0

Since Work is equivalent to Pressue x change in Volume (W = PdeltaV), no change in volume means no work gets done. When W = 0, the First Law reduces to U = Q.

In this case, internal energy depends entirely on heat. If heat is added to a system, Q is positive and internal energy goes up. If heat is removed from the system, Q is negative and internal energy goes down. Typical examples involve reactions in a rigid containers, where changes in volume are not possible.