Second Virial Coefficient, numerically

[Selveste]

Since the integrand is spherically symmetric I use spherical coordinates

$$\int d\vec{r} = \Omega_d\int_{0}^{\infty}dr r^{d-1}$$

where $\Omega_d$ is the solid angle i $d$ dimensions. Since I am to plot as a function of $1/\beta U_0 (= k_bT/U_0)$, I introduce a new varible $\Theta = 1/\beta U_0 = k_bT/U_0$. This gives

$$B_2(T) = \frac{1}{2} \Omega_d \Bigg( \int_0^R dr r^{d-1} + \int_R^{\infty} dr r^{d-1}(1-e^{\frac{1}{\Theta}(\frac{R}{r})^{\alpha}}) \Bigg)$$

Now I wonder if this is correct. And how to procede to compute the integrals numerically. Here $R$ is not given. Should I just set it to some value? I thought about integrating with respect to some varible $Rr$ or $r/R$, but I can't seem to get rid of $R$.

 

[member 553083]

Yes, this is correct. All that remains is to calculate the two integrals numerically. To do that, you can use a numerical integration technique such as the trapezoidal or Simpson's rule, or you can use a computer algebra system such as Mathematica or Maple. The first integral will be a simple one-dimensional integral, while the second integral will require more work since it involves an exponential function.When calculating the integrals, you should set R to some value in order to obtain a result. It is best to choose a value of R that is appropriate for the given problem. In this case, it might be reasonable to set R equal to the average radius of the particles.