# The1stLawofThermodynamicsLesson4

**THE JOULE - THOMSON EXPERIMENT **

The Joule experiment (see Fig. 1.5) shows that ΔT = 0 when an ideal gas expands under the condition w = O. By careful measurements Joule and Thomson found a small temperature change, usually negative, when a real gas expanded adiabatically through a porous plug. The experimental arrangement is shown in Fig. 4.1.

Fig. 4.1. The Joule - Thomson experiment .Adiabatic expansion of a real gas through a porous plug. (a) Initial state. (b) Final state.
The pressures P_{I} and P_{2} are kept constant during the expansion. The work supplied to the gas on the left hand side of the porous plug is P_{1}V_{I},and the work carried out by the gas on the right hand
side of the porous plug is P_{2}V_{2}. The net work received by the gas is:

**w = P _{1}V_{1}-P_{2}V_{2}** (4.2)

The process is adiabatic, q = O. From the first law we have:

ΔU =U_{2}-U_{1}= w = P_{1}V_{1}-P_{2}V_{2}

which gives:

**U _{2}+P_{2}V_{2}=U_{1}+P_{1}V_{1}**

From the definition of enthalpy (eq. (4.11)) we obtain:

** H _{2} = H_{1} ** (4.3)

This means that the enthalpy is constant during the expansion.
Joule and Thomson observed that the pressure change, ΔP = P_{2} - P_{I}, gave a change in temperature, ΔT = T_{2} - T_{1}. For most gases at room temperature one observes a positive ratio ΔT/ΔP.
The differential:

is called the Joule - Thomson coefficient. The total differential, dH, can be expressed by the Joule ¬Thomson coeffisient. For the function H = f(P, 1), the total differential is expressed by eq. (2.12):

The last term is equal to C_{p}dT (compare eq. (3.2)). For the Joule - Thomson experiment dH = 0, see
eq. (4.3). Dividing eq. (2.12) with dT for dH = 0 we obtain:

or:

Thus the change in enthalpy with changes in P and T is equal to:

**dH = - CpμdP + C _{p}dT** (4.6)

This equation is used in calculations of the enthalpy of real gases at high pressures when experimental values are known for the Joule - Thomson coefficient. Joule - Thomson expansion is important in refrigeration and in the liquefaction machine for condensing gases to liquids at very low temperatures.

** FUNDAMENTAL EQUATIONS**

The first law of thermodynamics:

**dU=dq+dw**(4.7) (4.7)

For a cyclic process:

The reversible pressure - volume work:

Isothermal reversible pressure - volume work supplied to an ideal gas:

w= -nRT ln (V(4.10)_{2}/V_{1})

Definition of enthalpy:

**H= u+pv** (4.11)

Heat capacities:

**C _{v}= (∂U/∂T)_{y}** (4.12)

**C _{p} = (∂H/∂T)_{p}** (4.13)