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.