In Problems 50 through 53 , the initial state of $1.00 \text{~mol}$ of a dilute gas is $P_1=3.00 \text{~atm}, V_1=1.00 \text{~L}$, and $E_{\text {int } 1}=456 \text{~J}$, and its final state is $P_2=2.00 \text{~atm}, V_2=3.00 \text{~L}$, and $E_{\text {int } 2}=912 \text{~J}$.

The gas is allowed to expand isothermally until it reaches its final volume and its pressure is $1.00 \text{~atm}$. It is then heated at constant volume until it reaches its final pressure. $(a)$ Illustrate this process on a $P V$ diagram and calculate the work done by the gas.

How many atoms of carbon are in one molecule of acetate ?

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. There is a relative extremum at each critical number.

Steam in a piston-cylinder assembly undergoes a polytropic process, with $n=2$, from an initial state where $p_1=$ $500 \ \mathrm{lbf} /$ in. $^2, v_1=1.701\ \mathrm{ft}^3 / \mathrm{lb}, u_1=1363.3\ \mathrm{Btu} / \mathrm{lb}$ to a final state where $u_2=990.58\ \mathrm{Btu} / \mathrm{lb}$. During the process, there is a heat transfer from the steam of magnitude $342.9$ Btu. The mass of steam is $1.2\ \mathrm{lb}$. Neglecting changes in kinetic and potential energy, find the work, in Btu, and the final specific volume, in $\mathrm{ft}^3 / \mathrm{lb}$.

In this exercise, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.

If $y=a x+b$, then $\Delta y / \Delta x=d y / d x$.

Do beavers benefit beetles? Researchers laid out 23 circular plots, each 4 meters in diameter, in an area where beavers were cutting down cottonwood trees. In each plot, they counted the number of stumps from trees cut by beavers and the number of clusters of beetle larvae. Ecologists think that the new sprouts from stumps are more tender than other cottonwood growth so that beetles prefer them. If so, more stumps should produce more beetle larvae. Here are the data.

$$\begin{array}{|l|c|c|c|c|c|c|c|c|c|c|c|c|} \hline \text { Stumps } & 2 & 2 & 1 & 3 & 3 & 4 & 3 & 1 & 2 & 5 & 1 & 3 \\ \hline \begin{array}{l} \text { Beetle } \\ \text { larvae } \end{array} & 10 & 30 & 12 & 24 & 36 & 40 & 43 & 11 & 27 & 56 & 18 & 40 \\ \hline \text { Stumps } & 2 & 1 & 2 & 2 & 1 & 1 & 4 & 1 & 2 & 1 & 4 & \\ \hline \begin{array}{l} \text { Beetle } \\ \text { larvae } \end{array} & 25 & 8 & 21 & 14 & 16 & 6 & 54 & 9 & 13 & 14 & 50 & \\ \hline \end{array}$$

Use technology to calculate the equation of the least-squares regression line relating y= number of larvae to x= number of stumps.

All atoms have protons, which carry a(n) _ charge.

Underline each verb in the following sentences.

Organs inside these beetles and their larvae emit a glow.

Plot the points in the Cartesian plane.

$$(0,0),(3,1),(-2,4),(1,-1)$$

A "canned heat" product used to warm chafing dishes consists of a homogeneous mixture of ethanol $\left(\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}\right)$ and paraffin that has an average formula of $\mathrm{C}_{24} \mathrm{H}_{50}$. What mass of $\mathrm{C}_2 \mathrm{H}_5 \mathrm{OH}$ should be added to $620 \mathrm{~kg}$ of the paraffin in formulating the mixture if the vapor pressure of ethanol at $35^{\circ} \mathrm{C}$ over the mixture is to be 8 torr? The vapor pressure of pure ethanol at $35^{\circ} \mathrm{C}$ is 100 torr.

Calculate the capacitive reactance, $X_\text{C}$, for the following capacitance and frequency values: a. $C=0.47\space\mu\text{F}$, $f=1\text{ kHz}$. b. $C=100\space\mu\text{F}$, $f=120\text{ Hz}$. c. $C=250\text{ pF}$, $f=1\text{ MHz}$. d. $C=0.0022\space\mu\text{F}$, $f=50\text{ kHz}$.

As Logan slows his bike, what happens to the EMF produced by his bike’s generator? Use the term armature in your explanation.

Factor the trinomial.

$$x^2 - 8x + 15$$

Marin Company’s general ledger indicates a cash balance of $22,340 as of September 30, 2021. Early in October Marin received a bank statement indicating that during September Marin had an NSF check of$1,500 returned to a customer and incurred service charges of $45. Marin also learned it had incorrectly recorded a check received from a customer on September 15 as$500 when in fact the check was for $550. Calculate Marin’s correct September 30, 2021, cash balance.