Final answer:
The dimensions of the constant 'a' in the van der Waals equation of state is [M⁻¹ Lµ T⁻²]. The constant 'b' has the dimensions of volume, [L³], which is directly subtracted from the molar volume 'v' in the equation.
Step-by-step explanation:
The question regarding the dimensions of the constants 'a' and 'b' in the van der Waals equation of state involves understanding dimensional analysis within the realm of thermodynamics in physics. The van der Waals equation is represented as (P + a/v²)(v - b) = RT, where P is the pressure, v is the molar volume, R is the gas constant, and T is the temperature. The constants 'a' and 'b' are empirical constants that correct the ideal gas law for the volume occupied by the gas molecules and the intermolecular forces between them, respectively.
To find the dimensions of 'a', we can look at the term a/v². The pressure P has dimensions of [M L⁻¹ T⁻²], so to add 'a' to pressure, 'a' must have dimensions that, when divided by volume squared, equal the dimensions of pressure: 'a': [M⁻¹ Lµ T⁻²]. Thus, option (a) [M⁻¹ Lµ T⁻²] is the correct dimension for constant 'a'.
As for constant 'b', its dimension comes directly from volume (as it is subtracted from the molar volume 'v'), so 'b': [L³]. However, this dimension is not among the options given as it's already known - it doesn't need to be derived from the equation.