Question

4) Given that $(\alpha /p\beta )=\alpha z/K_B\theta$ where p is a pressure, z is a distance, K_B is Boltzman constant and $\theta$ is temperature, the dimensions of $\beta are(usefulformulaEnergy=K_B\times tempreture)$.

Answer 1

Dear student,

$$\begin{gathered} \frac{\alpha }{p\beta }isdimensionless\sin ceitisunderthe\ln function. \\ \frac{\alpha z}{K\theta }isalsodimensionless\sin ceitisequalto\ln thatisdimensionless. \\ hence \\ \frac{1}{p\beta }=\frac{z}{K\theta }dimensionallyequal \\ \beta =\frac{\theta K}{Pz}=\frac{\left[K\right]\left[M{L}^2{T}^{-2}{K}^{-1}\right]}{\left[M{L}^{-1}{T}^{-2}\right]\left[L\right]}=\left[L^2\right] \\ Regards \end{gathered}$$