Numerically evaluate for one mole of methane acting as a van der Waals gas at (a) T = 298 K and V = 25.0 L and (b) T = 1000 K and V = 250.0 L. Comment on which set of conditions yields a number closer to that predicted by the
(a)
Interpretation:
Concept introduction:
The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.
Answer to Problem 1.51E
Explanation of Solution
The ideal gas equation can be represented as;
PV = nRT …(1)
Notably, the van Waals equation improves the ideal gas law by adding two significant terms in the ideal gas equation: one term is to account for the volume of the gas molecules and another term is introduced for the attractive forces between them. The non-ideal gas equation represented as;
In the above equation,
‘a’ and ‘b’ are called as van der Waals constants
Rearranging equation (2) we get pressure p of real gas as,
On differentiation with respect to volume V, at constant T, n we get
Given for methane,
Number of moles = n = 1 mole
Temperature of gas = T = 298 K
Volume of gas = V = 25.0 L
Value of constant ‘a’ for methane = 2.253atmL2/mol2
Value of constant ‘b’ for methane = 0.0428 L/mol
Substituting the values in equation (4), we get,
Besides, differentiating the equation (1) for ideal gas with respect to volume V, we get
Substituting the given parameters for methane in equation (5), we get for ideal gas
25.0 L is calculated as -0.0395 atm/L
(b)
Interpretation:
T = 1000 K and V = 250.0 L.
Concept introduction:
The ideal gas law considered the molecules of a gas as point particles with perfectly elastic collisions among them in nature. Thus, a modification of the ideal gas equation was coined by Johannes D. van der Waals in 1873 to consider size of molecules and the interaction forces among them. It is generally denoted as the van der Waals equation of state.
Answer to Problem 1.51E
250.0 L is calculated as -0.0013 atm/L
Explanation of Solution
The ideal gas equation can be represented as;
PV = nRT … (1)
The non-ideal gas equation represented as;
In the above equation,
‘a’ and ‘b’ are called as van der Waals constants.
Rearranging equation (2) we get pressure P of real gas as,
On differentiation with respect to volume V, at constant T, n we get
Given for methane,
Number of moles = n = 1 mole
Temperature of gas = T = 1000 K
Volume of gas = V = 250.0 L
Value of constant ‘a’ for methane = 2.253atmL2/mol2
Value of constant ‘b’ for methane = 0.0428 L/mol
Substituting the values in equation (4), we get,
Besides, differentiating the equation (1) for ideal gas with respect to volume V, we get
Substituting the given parameters for methane in equation (5), we get for ideal gas,
Non-ideal gas value is in close proximity to ideal gas values.
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Physical Chemistry