
Calculate the Boyle temperatures for carbon dioxide, oxygen, and nitrogen using the van der Waals constants in Table 1.6. How close do they come to the experimental values from Table 1.5?

Interpretation:
The Boyle temperatures for carbon dioxide, oxygen, and nitrogen using the van der Waals constants in Table 1.6. is to be calculated. And is to be compared to the experimental values from Table 1.5.
Concept introduction:
The Boyle temperature is defined as the temperature for which the second virial coefficient, becomes zero. In other words, the Boyle temperature is the temperature at which a non-ideal gas behaves like an ideal gas. In this temperature the attractive and repulsive forces acting on the gas molecules balance each out.
Answer to Problem 1.38E
The Boyle temperature of gas are as follows;
Entry | Gas | TB (K)(calculated values) | TB (K)(experimental values) | a (atm L2 / mol2) | B (L / mol) |
1 | Carbon dioxide | 1026 | 713 | 3.592 | 0.04267 |
2 | Oxygen | 521 | 405 | 1.360 | 0.03183 |
3 | Nitrogen | 433 | 327 | 1.390 | 0.03913 |
The experimental values of Boyle temperature (TB) are compared with calculated values and found to be in low.
Explanation of Solution
The Boyle temperature (TB) is defined as the temperature for which the second virial coefficient, becomes zero. In other words, the Boyle temperature is the temperature at which a non-ideal gas behaves like an ideal gas. In this temperature the attractive and repulsive forces acting on the gas molecules balance each out. Thus ‘TB’ is given by the expression
‘a’ and ‘b’ are called as van der Waals constants and ‘a’ represents the pressure correction and it is related to the magnitude and strength of the interactions between gas particles. Similarly, ‘b’ describes the volume correction and it is having relationship to the size of the gas particles.
Since, the Boyle temperature is the temperature for which second virial coefficient becomes 0. Moreover, it is at this temperature that the forces of attraction and repulsion acting on the gas molecules balance out. Higher order be virial coefficients are smaller than the second virial coefficient, the gas likes to behave as an ideal gas over a entire range of pressures. At low pressures,
the equation transforms as,
Where, Z = Compressibility factor.
TB of carbon dioxide
Knowing a and b values of CO2 from table 1.6 TB of CO2 is calculated as follows;
R = 0.0823 L. atm / K. mol
TB of oxygen:
Knowing a and b values of oxygen from table 1.6 TB of oxygen is calculated as follows;
TB of nitrogen:
Knowing a and b values of nitrogen from table 1.6 TB of nitrogen is calculated as follows;
Moreover, the experimental Boyle temperature (table 1.5) values are compared with calculated values and the experimental values are found to be low.
Thus, the Boyle temperatures for carbon dioxide, oxygen, and nitrogen using the van der Waals constants in Table 1.6. is calculated. And compared to the experimental values from Table 1.5.
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