When 25.0 g NaCl and 29.0 g H2SO4 are mixed and react according to the equation below. [on hold]

. When 25.0 g NaCl and 29.0 g H2SO4 are mixed and react according to the equation below.

2NaCl + H2SO4 = Na2SO4 + 2HCl

a) Which is the limiting reagent?
b) Calculate the theoretical mole of sodium sulfate.
c) Calculate the theoretical yield of sodium sulfate.
d) Given 26.50g of sodium sulfate is collected. Calculate the percent yield of the reaction.

Is there an extension of Wald’s Equation to the expectation of a product of random variables?

So Wald’s Equation states that for a real-values, independent and identically distributed sequence of random variables $(X_{n})_{ninmathbb{N}}$ and a nonnegative integer $N$, which is independent of the sequence, we have that:

$mathbb{E}[X_{1}+dots+X_{N}]=mathbb{E}[N]mathbb{E}[X_{1}]$

Under the assumption that both have finite expectation.

My question is if there exists some extension or similar identity that can be applied to the same type of problem except then with multiplication. I’m not sure if this simply follows from independence in most cases though. For example, given the same setting as earlier, do we have that:

$mathbb{E}[prod^{N}_{i=1}X_{i}]=prod^{N}_{i=1}mathbb{E}[X_{i}]$

Or is the outcome something comparable to the Wald’s Equation?

Any help is appreciated!

solving an equation for two lists of values sequentially

I’m very new to this. My goal is to run a function for 10 different pairs of values. The function looks like this:

f[groupsize_, solvedforq_] = 
Sum[Binomial[groupsize, t] Chop[solvedforq]^t (1 - Chop[solvedforq])^(groupsize - t), {t, 3, groupsize}]

The objects groupsize and solvedforq are lists, each containing 10 values. I would like to create a List or Table of outputs, where the first output is the solution for taking the first element of each list (i.e., the pair of values that I get when taking the first element of groupsize and the first element of solvedforq), the second output is the solution for taking the second element of each list, etc…

So I will ideally have a List or Table with 10 output values. I am ware for the command Map[], but I wouldn’t know how to use it in this more complex case. Thanks for all help!

How to find the smallest positive root of the following transcendental equation

What is the fastest way to find the smallest positive root of the following transcendental equation:

$$a + bcdot e^{-0.045 t} = n sin(t) – m cos(t)$$

eq = a + b E^(-0.045 t) == n Sin[t] - m Cos[t];

where
$a,b,n,m$ are some real constants.

for instance I tried :

eq = 11 + 5  E^(-0.045 t) == 0.03 Sin[t] - 1.2 Cos[t];

sol = FindRoot[eq, {t, 1}]

enter image description here

There is an answer but does it mean this is the smallest positive root? ))

I also tried Wolfram Alpha but there is no answer…

enter image description here

Method to find PDE equation coefficient satisfying mean solution?

What is the best approach to go about solving a PDE problem of the type

begin{equation}
k^3Delta u + knabla u = 0, ,\
u=g; text{on}; Gamma_D, ,\
mean(u) = u_text{mean}
end{equation}

where one wants to find for which positive constant coefficient $k>0$ the mean of the solution $u$ fulfills a prescribed value $u_text{mean}$?

discretizing the convection-diffusion equation using finite difference method

I am new to to the finite difference method and i want to understand how convection-diffusion equation is descritized in 2-D using central diffrences

$nabla .(rho vec{v} Phi)=nabla.(Gamma vec{nabla} Phi)$

I believe the descritization would yield :

$ frac{Phi_{i+1,j} – Phi_{i-1,j}}{2h}v_x+frac{Phi_{i,j+1} – Phi_{i,j-1}}{2h}v_y =frac{Gamma}{rho} frac{Phi_{i+1,j} + Phi_{i-1,j}+Phi_{i,j+1} + Phi_{i,j-1}-4Phi_{i,j}}{h^2} $

i hope i’ve got it right, now i believe this is the case for a problem with constant velocity,density etc
what i am i want to know is how can we do the same for a problem, with, a non uniform velocity for example.

edit : okay, so i worked a little more on the problem and i would really appreciate if someone could confirm or point out how the result is wrong, this is what i ended up with for a non uniform velocity:

$ frac{Phi_{i+1,j} – Phi_{i-1,j}}{2h}v_{x_{i,j}}+frac{Phi_{i,j+1} – Phi_{i,j-1}}{2h}v_{y_{i,j}} +Phi_{i,j}frac{(v_{x_{i+1,j}} -v_{x_{i-1,j}}) + (v_{y_{i,j+1}} -v_{y_{i,j-1}})}{2h} =frac{Gamma}{rho} frac{Phi_{i+1,j} + Phi_{i-1,j}+Phi_{i,j+1} + Phi_{i,j-1}-4Phi_{i,j}}{h^2} $

Velocity-Stress formulation of Elastodynamics/Wave Equation for beginner

I’m used to displacement forumlation of elastic wave equation:
$$
nabla cdot sigma (u) + F = rho ddot{ u }
$$
where $u$ is the primary variable. Recenty I started experimenting with DG and in almost every paper the stress-velocity formulation is used as a “conservative or divergence form”. What’s special about this formulation? I can’t figure out why they use Inerior Penalty with displacement equation and other fluxes like Lax-Friedrich’s with a system of $v, sigma$.
I would be thankful for any kind of paper, book or resource.

Tsiolkovysky’s Rocket Equation

As we know the rocket equation,

$$ Delta v = v_e lnleft(frac{m_i}{m_f}right) = I_{sp} g lnleft(frac{m_i}{m_f}right) $$

So do $I_{sp}$ and the mass ratio have an inverse relation, or is it that $I_{sp}$ is inversely related to natural log of the mass ratio?

Does it mean that to have logarithmic relation implies in the short interval (early phase) the difference is very large?