The First Law of Thermodynamics is a statement of conservation of energy. If dq is the heating of a parcel of air and dw is the work done by the parcel of air, then the change in internal energy of the air, du is given by:

du = dq - dw .

Every term in the equation has the units of energy (Joules or Nt m, or kg m²s^-2).

Consider the expansion of a parcel of air, say against a piston. We know that work is force x distance, so if the piston with area A is pushed out by the air, then the amount of work done as the piston is moved a distance dx is:

dW = p A dx = p dV.

We can integrate both sides to get:

W = ò p dV.

If we know the functional form of p on V, then we can integrate this expression. If we look at a unit mass of material (divide everything by m), then

dw = p da and dq = du + p da

2. Joules Law.

Joule's Law states that the internal energy of an ideal gas does not depend on volume if no heating or work occurs. All this means is that molecules are sufficiently far apart that we can move them closer together and their attraction and repulsion of each other (potential energy) does not change.
 
 

C. Specific heats.

As you heat any material, its temperature rises as its internal energy rises. In a gas, this energy goes into making the molecules move faster and rotate more rapidly. The specific heat of the material is defined as the amount of heat, dq, required to raise a unit mass of material a unit change in temperature, dT. If the volume is held constant, the specific heat, constant volume (c_v) is given by the expression:

c_v = (dq/dT)_v = (du/dT)_v = du/dT (because of Joule's Law)

 
 We can equate dq and du because the volume does not change and the work done is thus 0.

Now, we can also define a specific heat for conditions in which the pressure is kept constant:

c_p = (dq/dT)_p

In this case, the material is allowed to expand to maintain the pressure. Substituting these expressions for specific heat into the First Law of Thermodynamics, we have:

dq = du + dw = c_v dT + d(p a ) - a dp = (c_v + R) dT - a dp = (c_v + R) dT at constant pressure.

Thus, c_p = c_v + R.

Enthalpy is defined as :

h = u + p a .

It has units of energy. With a little mathematics, we can see that the specific heat, constant pressure is equal to:

c_p = dh/dT or h = c_p T after integration.

Enthalpy is one of the most important quantities in thermodynamics. In an air mass that is not heated or cooled, the quantity h + F is constant, where F is the geopotential.