Write the equilibrium constant expressions for these reactions:

(a) $3 \mathrm{O}_2(g) \rightleftharpoons 2 \mathrm{O}_3(g)$
(b) $\mathrm{H}_2 \mathrm{O}(g) \rightleftharpoons \mathrm{H}_2 \mathrm{O}(l)$
(c) $\mathrm{MgCO}_3(s) \rightleftharpoons \mathrm{MgO}(s)+\mathrm{CO}_2(g)$
(d) $2 \mathrm{~Bi}^{3+}(a q)+3 \mathrm{~H}_2 \mathrm{~S}(a q) \rightleftharpoons \mathrm{Bi}_2 \mathrm{S}_3(s)+6 \mathrm{~H}^{+}(a q)$

In this exercise, use the Root Test to determine the convergence or divergence of the series.

$$\displaystyle\sum_{n=1}^{\infty}\left(\frac{n}{2 n+1}\right)^n$$

How do economies of scale apply to physicians?

Consider a harmonic oscillator with mass m=0.100 kg and k = 50 N/m. What is the separation ∆E between energy levels in this harmonic oscillator?

Choose the correct answer.

(a) Who, (b) Whom would you like to speak with about your idea for a new product?

Write each expression in the standard form a + bi. $(1-\sqrt{5} i)^{8}$

For a neuron to propagate a signal in a nerve impulse within itself down its length, ____ is necessary to happen.

a. postsynaptic wave

b. synaptic gap

c. depolarization wave

d. repolarization wave

Integrate each of the given finctions.

$$\displaystyle\int \frac{d x}{e^x+e^{-x}}$$

How can you rewrite the expression $(8-5 i)^{2}$ in the form a+bi? F. 39+80i G. 39-80i H. 89+80i I. 89-80i

Solve the following problem using the MATLAB environment.

There are mechanical, electrical, and chemical systems that are described by the same mathematics as second-order, forced, damped harmonic motion. The resulting differential equation obtained after applying a force balance or conservation of momentum is of the form:

$$m \frac{d^2 x}{d t^2}+\gamma \frac{d x}{d t}+k x=A_0 \sin (\omega t)$$

where $x$ is the displacement, $t$ is time, $m$ is the mass, $\gamma$ is the damping coefficient, $k$ is the restoring force (spring) constant, $A_0$ is the amplitude of the driving force, and $\omega$ is the frequency of the driving force. (a.) Determine the order of the ODE. Is it linear, nonlinear, homogeneous or non homogeneous? (b.) Find the homogeneous solution of the ODE by hand. (c.) Find the particular solution of the ODE by hand. Find $x(t)$ after a long time $(t \rightarrow \infty)$. This is sometimes called the "steady state" response, even though it is actually time varying. (d.) Using MATLAB, plot the maximum amplitude of $x(t)$ from the steady state response as a function of the excitation frequency $\omega(0 \leq \omega \leq 5 \mathrm{rad} / \mathrm{s})$ for $A_0=1 \mathrm{~N}, k=1 \mathrm{~N} / \mathrm{m}$, and $m=1 \mathrm{~kg}$, for three values of $\gamma: \gamma=0.5, \gamma=1.0$, and $\gamma=2.0 \mathrm{~N}-\mathrm{s} / \mathrm{m}$ (three plots on the same figure). Discuss the results. What happens at $\omega=\sqrt{k / m}$ when $\gamma=0$ ?

Explain how the statement "You dump it, you drink it!" could apply to used motor oil.

Derive the following features of the gamma $(r, \lambda)$ distribution for all positive $r$ :

a) For $r \geq 1$ the mode (i.e., the value that maximizes the density) is $(r-1) / \lambda$. What if $0<r<1$ ?

b) For $k>0$, the $k$ th moment of $T$ with gamma $(r, \lambda)$ distribution is

$$E\left(T^k\right)=\frac{1}{\lambda^k} \frac{\Gamma(r+k)}{\Gamma(r)}$$

In particular $E(T)=r / \lambda$.

c) $S D(T)=\sqrt{r} / \lambda$ and Skewness $(T)=2 / \sqrt{r}$.

Write each expression in the standard form a + bi. $\left(\frac{\sqrt{3}}{2}-\frac{1}{2} i\right)^{2}$

Write each expression in the standard form a + bi. $\left[\sqrt{3}\left(\cos \frac{5 \pi}{18}+i \sin \frac{5 \pi}{18}\right)\right]^{6}$