The ideal gas law states that PVₘ = RT, where Vₘ = V/n is the volume per mole of gas. However, any real gas will exhibit deviations from the ideal gas law, described by the virial expansion,
PVm=RT(1+VmB(T)+Vm2C(T)+...).
For gases with low density, the higher-order terms are negligible, so in this problem we will neglect all of the temperature-dependent terms in parentheses except for B(T)/Vₘ. The table below shows measurements of B for nitrogen gas (N₂) at atmospheric pressure, P = 1.01×10⁵Pa.
T (K)
B(cm³/mol)
100
−160
200
−35
300
−4.2
400
9.0
500
16.9
600
21.3
a. According to the ideal gas law, what is the value of Vₘ at temperatures 100 K, 300 K, and 600 K? Give your answers in SI units.
b. What is the percentage change in Vₘ at these temperatures if one accounts for B(T)?
c. In 1910, van der Waals was awarded the Nobel Prize for formulating the equation
(P+Vm2a)(Vm−b)=RT
which accurately describes many real gases. According to this equation, what is the form of B(T)? You may assume that b ≪ Vₘ.
d. Using the data above, extract the values of a and b. Give your answers in SI units.
e. In this problem, we have neglected terms in the virial expansion beyond B(T), which is a good approximation as long as the volume correction due to B(T) itself is small. Assuming the van der Waals equation holds, numerically estimate the temperature range within which the volume correction due to B(T) is at most 10%, for nitrogen gas at atmospheric pressure.