To determine the unit of [tex]\(a\)[/tex] in the CGS (centimeter-gram-second) system, we need to analyze the provided equation:
[tex]\[
\left(p + \frac{a}{v^2}\right)(v - b) = \text{constant}
\][/tex]
First, let's consider the units involved in the terms of the equation.
1. Pressure ([tex]\(p\)[/tex]): The unit of pressure in the CGS system is dyn/cm² (dyne per square centimeter).
2. Volume ([tex]\(v\)[/tex]): The unit of volume in the CGS system is cm³ (cubic centimeter).
Next, let's break down the term [tex]\(\frac{a}{v^2}\)[/tex]:
For the term [tex]\(\frac{a}{v^2}\)[/tex] to be added to pressure [tex]\(p\)[/tex], it must have the same units as pressure. Thus, the unit of the term [tex]\(\frac{a}{v^2}\)[/tex] must also be dyn/cm².
Rewriting the term [tex]\(\frac{a}{v^2}\)[/tex] in terms of units, we have:
[tex]\[
\frac{a}{(\text{cm}^3)^2}
\][/tex]
Since this must be equal to dyn/cm², we equate the units:
[tex]\[
\frac{[a]}{\text{cm}^6} = \text{dyn/cm}^2
\][/tex]
Solving for the unit of [tex]\(a\)[/tex], we multiply both sides by cm⁶:
[tex]\[
[a] = \text{dyn} \cdot \text{cm}^6
\][/tex]
This clarifies that the unit of [tex]\(a\)[/tex] must be such that when divided by volume squared (cm⁶ in this case), it yields the unit of pressure (dyn/cm²). Therefore, in CGS units, the unit of [tex]\(a\)[/tex] is:
[tex]\[
\text{dyn} \cdot \text{cm}^4
\][/tex]
Thus, the unit of [tex]\(a\)[/tex] in the CGS system is:
[tex]\[
\text{dyn} \cdot \text{cm}^4
\][/tex]
