Tris(hydroxymethyl)aminomethane, commonly called TRIS or Trizma, is often used as a buffer in biochemical studies. Its buffering range is from pH 7 to $9 ,$ and $K _ { b }$ is $1.19 \times 10 ^ { - 6 }$ for the reaction

$$\left( \mathrm { HOCH } _ { 2 } \right) _ { 3 } \mathrm { CNH } _ { 2 } ( a q ) + \mathrm { H } _ { 2 } \mathrm { O } ( l ) \rightleftharpoons \left( \mathrm { HOCH } _ { 2 } \right) _ { 3 } \mathrm { CNH } _ { 3 } + ( a q ) + \mathrm { OH } ^ { - } ( a q )\\ {\begin{array} {l}{\text{TRIS} \quad \quad \quad \quad \quad \quad \quad \quad \text{TRISH}^+} \end{array}}$$

What is the optimum pH for TRIS buffers?

Find the unit tangent vector to the curve at the specified value of the parameter. r(t) = 3ti - ln tj, t=e

When triangle BUG is graphed in the coordinate plane, the slope of $\overline{B U}=-18$, the slope of $\overline{G U}=\frac{1}{2}$, and the slope of $\overline{B G}=-2$. Is triangle BUG a right triangle? Explain your reasoning.

The diameter of a neutral helium atom is about

$$1\times10^2\;\mathrm{pm}.$$

Suppose that we could line up helium atoms side by side in contact with one another. Approximately how many atoms would it take to make the distance 1 in from end to end?

The first six terms of the Fibonacci sequence are: 1, 1, 2, 3, 5, and 8. Determine the 11th and 12th terms of the Fibonacci sequence.

Determine whether the statements are true or false. Every nonzero subspace of $R^n$ has a unique basis.

The speed of sound is approximately

$$330 \mathrm { ms } ^ { - 1 }$$

. A storm is 3 km away. Approximately how much later after seeing lightning will thunder be heard? A. 0.1 s B. 1 s C. 3 s D. 10 s

Determine the $\ce{pH}$ of a 0.500 M $\ce{HNO2}$ solution. $K_a$ of $\ce{HNO2}$ is $4.6 \times 10^{−4}$.

Magnetic-Optical Corporation offers a variety of share-based compensation plans to employees. Under its restricted stock award plan, the company on January 1, 2016, granted 4 million of its $1 par common shares to various division managers. The shares are subject to forfeiture if employment is terminated within three years. The common shares have a market price of$22.50 per share on the grant date. Required: 1. Determine the total compensation cost pertaining to the restricted shares. 2. Prepare the appropriate journal entry to record the award of restricted shares on January 1, 2016. 3. Prepare the appropriate journal entry to record compensation expense on December 31, 2016. 4. Suppose Magnetic-Optical expected a 10% forfeiture rate on the restricted shares prior to vesting. Determine the total compensation cost.

Determine whether each binomial is a difference of two squares. If so, factor. If not, explain.

$$x ^ { 6 } - 9$$

Suppose 50.0 ml of a 0.0500 M solution of Pb(NO3)2 is mixed with 40.0 ml of a 0.200 M solution of NaIO3 at 25°C. Calculate the [Pb2^+] and [IO^-3) when the mixture comes to equilibrium. At this temperature, Ksp for Pb(IO3)2 is 2.6×10^-13.

Calculate the length of each of the paths

$$\mathbf { x } ( t ) = \left( \cos 3 t , \sin 3 t , 2 t ^ { 3 / 2 } \right) , 0 \leq t \leq 2$$

Find the convolution $f ( t ) * g ( t )$.

$$f ( t ) = t ^ { 2 } , g ( t ) = \cos t$$

Find an estimated answer and an exact answer. a. 36.45 - 4.912 b. 4.2 × 0.9