Given the equation representing a reversible reaction: $\mathrm{NH}_{3}(g)+\mathrm{H}_{2} \mathrm{O}(\ell) \rightleftarrows \mathrm{NH}_{4} \ ^{+}(a q)+\mathrm{OH}^{-}(a q)$ According to one add-base theory, the reactant that donates an $\mathrm{H}^{+}$ ion in the forward reaction is: (1) $\mathrm{H}_{2} \mathrm{O}(\ell)$ (2) $\mathrm{NH}_{3}(g)$ (3) $\mathrm{NH}_{4}^{+}(a q)$ (4) $\mathrm{OH}^{-}(a q)$.

How does latitude relate to climate?

Assuming that air contains 78 percent N2, 21 percent O2, and 1 percent Ar, all by volume, how many molecules of each type of gas are present in 1.0 L of air at STP?

At what temperature will a gas sample occupy 100.0 L if it originally occupies 76.1 L at 89.5 °C? Assume constant $P$. (a) 276 °C (b) 118 °C (c) 203 °C (d) 68.1 °C (e) 99.6 °C

Rotate Ali The Alien 180°. Be sure to identify your center of rotation.

Explain the difference between a virtual image and a real image. Give an example of each type of image.

Prove that $\sqrt{2}+\sqrt{3}$ is irrational.

Let A be an $m\times n$ matrix of rank r, let I be a set of r row indices such that the corresponding rows of A are independent, and let J be a set of r column indices such that the corresponding columns of A are independent. Let M denote the $r\times r$ submatrix of A obtained by taking rows from I and column from J. Prove that M is invertible.

Compare the effects of short-term and long-term change in an ecosystem.

Prove that if A is a sy mmetric matrix whose spectral decomposition is $A=\lambda_{1} \mathbf{u}_{1} \mathbf{u}_{1}^{T}+\lambda_{2} \mathbf{u}_{2} \mathbf{u}_{2}^{T}+\cdots+\lambda_{n} \mathbf{u}_{n} \mathbf{u}_{n}^{T}$ then $f(A)=f\left(\lambda_{1}\right) \mathbf{u}_{1} \mathbf{u}_{1}^{T}+f\left(\lambda_{2}\right) \mathbf{u}_{2} \mathbf{u}_{2}^{T}+\cdots+f\left(\lambda_{n}\right) \mathbf{u}_{n} \mathbf{u}_{n}^{T}$.

Decide whether each line is parallel to the y-axis, the x-axis or neither: a. the line passing through the points P(1, 7) and Q(12, 7). b. the line passing through the points P(1, 7) and T(1, -3). c. the line passing through the points P(1, 7) and M(2, 5).

Evaluate each determinant.

$$\left| \begin{array} { r r } { \frac { 3 } { 8 } } & { \frac { 1 } { 2 } } \\ { - \frac { 1 } { 9 } } & { 5 } \end{array} \right|$$

What are the critical numbers of a function? What is the difference between critical numbers and critical points?

Wilma tells Terrarhce that since all of the expressions resulted in the same solution, any of the expressions can be used to determine the number of additional tiles needed to make more $n \times n$ designs. Terrance thinks that the employees need to use more;yalues in the expressions than just one to make this conclusion. Who Is correct? Explain your reasoning.