Virial Expansion Approximation of of Lennard Jones Potential

[Diracobama2181]

Homework Statement

Numerically evaluate the second viral coefficient of the Lennard-Jones Potential. The result should be expressed in the form of a dimensional constant times
a dimensionless function of the dimensionless variable $$ E_0 \beta$$

Relevant Equations

$$U=E_0((\frac{r_0}{r})^{12}-2(\frac{r_0}{r})^6)$$
$$B_2(T)=2\pi N\int_{0}^{\infty} (1-e^{-\beta E_0((\frac{r_0}{r})^{12}-2(\frac{r_0}{r})^6)})r^2dr$$

I get
$$B_2(T)=2\pi N\int_{0}^{\infty} (1-e^{-\beta E_0((\frac{r_0}{r})^{12}-2(\frac{r_0}{r})^6)})r^2dr$$
as the coefficient. I was just unsure how to evaluate it numerically from here. Any suggestions would be appreciated. Thank you.

 

[Abhishek11235]

Try expanding exponential function around 0 to few orders. Also,look at the graph of function (using any software) why you need Taylor's Expansion around 0