derive real gas equation
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van der Waals equation (or van der Waals equation of state; named after Johannes Diderik van der Waals) is based on plausible reasons that real gases do not follow the ideal gas law. The ideal gas law treats gas molecules as point particles that do not interact with each other but only with their containers. In other words, they do not take up any space, and are not attracted or repelled by other gas molecules and thus never lose kinetic energy during collisions. The ideal gas law states that volume (V) occupied by n moles of any gas has a pressure (P) at temperature (T) in kelvins. The relationship fothese variables is
PV = nRT,
where R is known as the gas constant, is called the ideal gas law or equation of state.
To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with {\displaystyle (V_{m}-b)}, where b is the volume that is occupied by one mole of the molecules. This leads to:[1]
{\displaystyle P(V_{m}-b)=RT}
The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact attract each other and that real gases are therefore more compressible than ideal gases. Van der Waals provided for intermolecular attraction by adding to the observed pressure P in the equation of state a term {\displaystyle a/V_{m}^{2}}, where a is a constant whose value depends on the gas. The van der Waals equation is therefore written as:[1]
{\displaystyle (P+a/V_{m}^{2})(V_{m}-b)=RT}
and can also be written as the equation below
{\displaystyle (P+an^{2}/V^{2})(V-nb)=nRT}
where Vm is the molar volume of the gas, R is the universal gas constant, T is temperature, Pis pressure, and V is volume. When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the van der Waals equation reduces to the ideal gas law, PVm=RT.[1]
It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified. It successfully approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the transitions between gas and liquid, in the range of p, V, and T where the liquid phaseand the gas phase are in equilibrium, the van der Waals equation fails to accurately model observed experimental behaviour, in particular that p is a constant function of V at given temperatures. As such, the van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Empirical corrections to address these predictive deficiencies have been inserted into the van der Waals model, e.g., by James Clerk Maxwell in his equal area rule, and related but distinct theoretical models, e.g., based on the principle of corresponding states, have been developed to achieve better fits to real fluid behaviour in equations of more comparable complexity.
PV = nRT,
where R is known as the gas constant, is called the ideal gas law or equation of state.
To account for the volume that a real gas molecule takes up, the van der Waals equation replaces V in the ideal gas law with {\displaystyle (V_{m}-b)}, where b is the volume that is occupied by one mole of the molecules. This leads to:[1]
{\displaystyle P(V_{m}-b)=RT}
The second modification made to the ideal gas law accounts for the fact that gas molecules do in fact attract each other and that real gases are therefore more compressible than ideal gases. Van der Waals provided for intermolecular attraction by adding to the observed pressure P in the equation of state a term {\displaystyle a/V_{m}^{2}}, where a is a constant whose value depends on the gas. The van der Waals equation is therefore written as:[1]
{\displaystyle (P+a/V_{m}^{2})(V_{m}-b)=RT}
and can also be written as the equation below
{\displaystyle (P+an^{2}/V^{2})(V-nb)=nRT}
where Vm is the molar volume of the gas, R is the universal gas constant, T is temperature, Pis pressure, and V is volume. When the molar volume Vm is large, b becomes negligible in comparison with Vm, a/Vm2 becomes negligible with respect to P, and the van der Waals equation reduces to the ideal gas law, PVm=RT.[1]
It is available via its traditional derivation (a mechanical equation of state), or via a derivation based in statistical thermodynamics, the latter of which provides the partition function of the system and allows thermodynamic functions to be specified. It successfully approximates the behavior of real fluids above their critical temperatures and is qualitatively reasonable for their liquid and low-pressure gaseous states at low temperatures. However, near the transitions between gas and liquid, in the range of p, V, and T where the liquid phaseand the gas phase are in equilibrium, the van der Waals equation fails to accurately model observed experimental behaviour, in particular that p is a constant function of V at given temperatures. As such, the van der Waals model is not useful only for calculations intended to predict real behavior in regions near the critical point. Empirical corrections to address these predictive deficiencies have been inserted into the van der Waals model, e.g., by James Clerk Maxwell in his equal area rule, and related but distinct theoretical models, e.g., based on the principle of corresponding states, have been developed to achieve better fits to real fluid behaviour in equations of more comparable complexity.
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