The equation of state of some gases can be expressed as (P + a/V²)(V − b) = RT. Here P is the pressure, V is t

Question

The equation of state of some gases can be expressed as (P + a/V²)(V − b) = RT. Here P is the pressure, V is the volume, T is the absolute temperature and a, b, R are constants. The dimensions of ‘a’ are:

Accepted Answer

According to the **Principle of Homogeneity of Dimensions**, only physical quantities having the same dimensions can be added to or subtracted from each other.

1.  In the term $(P + rac{a}{V^2})$, the quantity $rac{a}{V^2}$ must have the same dimensions as Pressure ($P$).
2.  **Dimensions of Pressure ($P$):** Force per unit area. 
    $$ [P] = rac{[Force]}{[Area]} = rac{[MLT^{-2}]}{[L^2]} = [ML^{-1}T^{-2}] $$
3.  **Dimensions of Volume ($V$):** $[L^3]$.
4.  **Calculate Dimensions of 'a':**
    $$ [rac{a}{V^2}] = [P] $$
    $$ [a] = [P] 	imes [V]^2 $$
    $$ [a] = [ML^{-1}T^{-2}] 	imes ([L^3])^2 $$
    $$ [a] = [ML^{-1}T^{-2}] 	imes [L^6] $$
    $$ [a] = [ML^5T^{-2}] $$

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