Question

In van der Wall equation

\([P=\frac{a}{V^2}[v-b]=RT;\)

P is pressure, V is volume, R is universal gas constant and T is temperature. The ratio of constants a/b is dimensionally equal to:

• A. \(\frac{P}{V}\)
• B. \(\frac{V}{P}\)
• C. PV
• D. PV³

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the van der Waals equation and determine the unit and dimensions of the constant a/b in the given context. The van der Waals equation is provided as:

P = \frac{a}{V^2}(V-b) = RT

Where:

• P is the pressure
• V is the volume
• R is the universal gas constant
• T is the temperature
• a and b are constants specific to the gas

For this problem, we need to find the dimensions of a/b. First, let's understand the dimensions of a and b in terms of basic quantities:

1. Dimension of a:

   From the equation P + \frac{a}{V^2} = \cdots, a is added to pressure, which means \frac{a}{V^2} must have the same dimensions as pressure. Hence,

   a = P \cdot V^2

   The dimensional formula for pressure (P) is [M][L^{-1}][T^{-2}], and for volume (V) is [L^3]. Therefore,

   The dimensional formula for a is [M][L][T^{-2}].

2. Dimension of b:

   b has the dimension of volume. Hence, its dimensional formula is [L^3].

Having determined these dimensions, the ratio \frac{a}{b} will have the following dimensions:

\frac{[M][L][T^{-2}]}{[L^3]} = [M][L^{-2}][T^{-2}]

This dimensional formula can be related to dimensions of pressure times volume:

The dimensions of PV are:

[M][L^{-1}][T^{-2}] \cdot [L^3] = [M][L^2][T^{-2}]

Therefore, the dimensional formula of \frac{a}{b} matches with that of PV.

Thus, the ratio a/b is dimensionally equal to PV.

Considering this calculation, the correct answer is:

PV