By T. Flynn
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And are functions of temperature only. , Tr ¼ T=Tc. T is the system temperature, and Tc is the critical temperature of the system component. Another useful generalized equation of state is the three-parameter corresponding states form employing the acentric factor o, Z ¼ f ðTr ; Pr ; oÞ ð2:23Þ where o is a tabulated parameter for a speciﬁc substance. 2. , can now be calculated for any process for any substance according to the following scheme. Step 1. Choose a pair of coordinates from P, T, and V, for example, P–T, T–V, or P–V.
So the total change in velocity is 2u. , molecular weight times the change in velocity, 2u). Thus, 1 n 1 Mn F¼ AuðMÞð2uÞ ¼ Au2 6 Vt 3 Vt and the pressure is F 1 Mn 2 P¼ ¼ u A 3 Vt The ideal gas law is P ¼ (n=V t) RT; thus, 1 Mu2 ¼ RT 3 or Mu2 3 ¼ RT 2 2 where Mu2=2 is the average kinetic energy per mole. Thus, we have shown that for an ideal gas the kinetic energy is directly proportional to the absolute temperature. Or, put another way, temperature is a direct measure of kinetic energy.
Thus, 1 n 1 Mn F¼ AuðMÞð2uÞ ¼ Au2 6 Vt 3 Vt and the pressure is F 1 Mn 2 P¼ ¼ u A 3 Vt The ideal gas law is P ¼ (n=V t) RT; thus, 1 Mu2 ¼ RT 3 or Mu2 3 ¼ RT 2 2 where Mu2=2 is the average kinetic energy per mole. Thus, we have shown that for an ideal gas the kinetic energy is directly proportional to the absolute temperature. Or, put another way, temperature is a direct measure of kinetic energy. Notice also that pressure is equal to kinetic energy per unit volume. This short derivation also provides some insight into the universal gas constant R.
Cyrogenic Engineering by T. Flynn