Integrating Non-Constant Charge Density in a Volume: Tips and Tricks

In summary, the problem is to find the total charge inside a given volume with a charge density function that includes both cartesian and cylindrical variables. The solution involves integrating with respect to one variable at a time while holding the others constant, and the final answer will be in terms of the constant rho. Trying to convert rho to a constant using substitution or integration by parts does not work in this case.
  • #1
stargoo
3
0

Homework Statement


Find the total charge inside the volume indicated:
[tex]\rho_v=10z^2\rho^{-0.1x}\sin(y\pi) for -1\leq x\leq 2,0\leq y\leq 1,3\leq z \leq3.6[/tex]

I know I have to integrate over the volume [tex]dxdydz[tex], but [tex]\rho^{-.1x}[tex] just keeps giving me a problem. Is there a substitution that I'm missing here. Integrating by parts just gives a more complicated integral. I've tried converting [tex]\rho[tex] to [tex]\sqrt{x^2+y^2}[tex], but that doesn't help either. Any help would be appreciated.
 
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  • #2
got this from google [integral] b^{x} dx = b^{x} / ln(b) + C

Simple: look in integral tables :P
 
  • #3
that would be all well and good, except that rho is not a constant in this case. The charge density in the stated problem is a function of cartesian AND cylindrical variables. Hence, my problem.
 
  • #4
oops, I couldn't read properly the question..

It seems a real big mess to me ><
 
  • #5
thanks for trying, though!
 
  • #6
Are you sure rho is not a constant? The way I read the problem is rho(sub-v) represents some total charge (although that notation would seem a bit unconventional) defined as a function of x, y, z, and some undeclared (but constant) rho. From that point of view, the integral really isn't as ugly as it looks. The triple integral is performed by integrating with respect to one variable at a time, holding the others constant. Your final answer will just be in terms of rho.
If rho were variable, then you have a variable defined as a function of its anti-derivative, which is a differential equation, and you have a mess on your hands.
Good Luck!
 

1. What is meant by the "charge density problem"?

The charge density problem is the discrepancy between the predicted and observed values of the electron density in molecules. It is a fundamental issue in quantum chemistry and refers to the difficulty in accurately calculating the distribution of electrons in a molecule.

2. Why is the charge density problem important in chemistry?

The charge density is a key property of a molecule that influences its reactivity, physical properties, and behavior. Therefore, accurately predicting the charge density is crucial for understanding and predicting the behavior of molecules in various chemical reactions and processes.

3. How is the charge density problem addressed in computational chemistry?

The charge density problem is addressed through various theoretical methods and approximations in computational chemistry. These include density functional theory (DFT), Hartree-Fock theory, and post-Hartree-Fock methods, which aim to improve the accuracy of electron density calculations.

4. What are some challenges in solving the charge density problem?

One of the main challenges in solving the charge density problem is the complexity of the quantum mechanical equations involved. These equations cannot be solved exactly, and thus, approximations must be made. Additionally, the accuracy of the calculations depends on the choice of the theoretical method and the basis set used.

5. How can the charge density problem impact drug discovery and material design?

The accurate prediction of charge density is essential in drug discovery and material design. It can provide insights into the binding interactions between molecules, their stability, and other important properties. Therefore, the charge density problem directly affects the success of these processes in the fields of pharmaceuticals and materials science.

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