Compression and expansion

Principle

To understand the concept of diabatic phenomena, first Boyle’s law is applied to the simple set-up of Fig. 1.

Fig. 1

The amount of gas in both compartments together is p·V_a + p·V_b which holds with the closed valve (at time zero, t₀) as well as after equilibration of the pressures (at time t) when the valve has been opened. So, it holds that:

p_{a, 0}·V_a + p_{b, 0}·V_{b =} p_t·(V_a + V_b). (1)

With given p_{a, 0}, p_{b, 0}, V_a and V_b at t_0, p_t can be calculated. When V_b = jV_a and p_{b, 0} = kp_{a, 0} then p_t becomes:

p_t = p_a(1+jk)/(1+j). (2)

However, this is only true when the temperature T in the whole system remains constant during the whole process of pressure equilibration, i.e. the process is isotherm or diabatic. When this is not the case, then the left compartment cools down and the right one heats up. For more info about this adiabatic process, see Adiabatic Compression and expansion.

Another condition is that the gas is ideal, i.e. the particles do not interact and they have no size. When pressures (so densities) are low, both conditions apply well and (1), which is actually based on Boyle’s law (see Gas Laws), can be used. When they do not hold, the Van der Waals corrections are necessary.

Application

Innumerous in science, technology and so indirectly in medicine, e.g. in breathing apparatus and especially in pulmonology.

The Van der Waals corrections are used for mass calculations of commercials gases in high-pressure tanks e.g. applied in medicine, especially with the expensive helium.

More Info

The Van der Waals correction has two constants a and b (Table 1). The correction factor “a” is needed for the interaction between the gas particles (attraction coefficient), and a correction factor “b” for the volume occupied by the gas particles.

Table 1.

Molecule,	Van der Waals constants
atom	a	b
or mixture	10^3J.m^3/kmol^2	10^-3m^3/kmol
He	3,5	22
H2	25	26
O2	140	31
N2	140	39
CO2	360	44
H2O	550	30.5
air	140	37.4

b of air interpolated from weighed b’s of O₂ and N₂

According to Van der Waals:

(p + an²/V_m²) (V_m-nb) = nRT, (3a)

where V_m the total volume and n the number of kmoles in the volume V. They increase and reduce pressure, respectively. Table 1 gives for some gases the numerical values of the two constants. When a and b are zero, then the van der Waals equation degenerates to Boyle's law. To calculate p, (3) can be rewritten:

p = nRT/(V_m-nb) - an²/V_m², (3b)

With normal temperatures and pressures of about 100 bar or more, the Van der Waals correction makes sense. For instance, a tank filled with air 300 bar and 290 K comprises 7.8% less. (assumed that a_air = 37.4x10^-3). At 200 bar the interaction effect dominates the particle-volume-effect, but at 300 bar the situation is reversed. The p-V diagram of Fig. 2 also illustrates the rather complex behavior of the correction. Since Boyle’s law is independent of the type of particles, the straight line of Boyle (log-log diagram) holds for both air and helium.

Fig. 2 Comparison of p/V relation according to the Law of Boyle and according to the Van der Waals equation for air and He. The curves are calculated for n = 0.1 kmol and T = 300 K (1E4 = 10^!4).

The Van der Waals curve of He shows already strong deviations (about 10%) from Boyle’s law at 100 bar, since the interaction effect is weak compared to the particle-volume-effect. The p-V curve approaches the straight Boyle line from above, but the air-curve first crosses the Boyle line and then approaches the line from below. For low pressures the Boyle line is the asymptote for the Van der Waals curves. The rather surprising behavior of the Van der Waals equation is due to the fact that it is a cubic equation in n and in V.

For very precise calculations in gas mixtures at pressures beyond 50 bar, this correction is even not precise enough. Then, the theory of virial coefficients, taking also into account the (first and higher order) interactions between the types of particles in a mixture (), is applied (see textbooks of physics).