The compressibility $\kappa$ of a substance is defined as the fractional change in volume of that substance for a given change in pressure:

$$\kappa=-\frac{1}{V} \frac{d V}{d P}$$

(a) Explain why the negative sign in this expression ensures $\kappa$ is always positive. (b) Show that if an ideal gas is compressed isothermally, its compressibility is given by $\kappa_{1}=1 / P$. (c) Show that if an ideal gas is compressed adiabatically, its compressibility is given by $\kappa_{2}=1 /(\gamma P) .$ Determine values for (d) $\kappa_{1}$ and (e) $\kappa_{2}$ for a monatomic ideal gas at a pressure of 2.00 atm.

Eight molecules have speeds of 3.00 km/s, 4.00 km/s, 5.80 km/s, 2.50 km/s, 3.60 km/s, 1.90 km/s, 3.80 km/s, and 6.60 km/s. Find (a) the average speed of the molecules and (b) the rms speed of the molecules.

Your sister, who is a realtor, is quite interested in your studies in physics. Part of her job in selling properties involves being aware of the details of heating systems for the houses. She comes to you one day and says that a client told her the following: "If you measure the internal energy in the air in a house and then turn up the thermostat to a higher temperature, the internal energy of the air in the house is exactly the same as it was at the lower temperature." She finds this hard to believe, since you have added energy to the air by running the furnace. Help her to figure out if this statement is true or not.

Air in a thundercloud expands as it rises. If its initial temperature is 300 K and no energy is lost by thermal conduction on expansion, what is its temperature when the initial volume has doubled?

Work each problem. Find a value of c so that $y=x^2-10x+c$ has exactly one x-intercept.

Let x=-4 and y=2. Evaluate each expression. $|-3 x+4 y|$

Discuss the validity of each statement. If the statement is always true, explain why. If not, give a counterexample. If f is a polynomial, then, as x approaches 0, the right-hand limit exists and is equal to the left-hand limit.

Write the augmented matrix for each system and give its dimension. Do not solve.

$$\begin{aligned} &4 x-2 y+3 z-4=0\\ &3 x+5 y+z-7=0\\ &5 x-y+4 z-7=0 \end{aligned}$$

Perform each operation. Write answers in standard form. $(4-3 i)^{2}$

In a study designed to investigate the effects of a strong magnetic field on the early development of mice, ten cages, each containing three 30-day-old albino female mice, were subjected for a period of 12 days to a magnetic field having an average strength of 80 Oe/cm. Thirty other mice, housed in ten similar cages, were not put in the magnetic field and served as controls. Listed in the table are the weight gains, in grams, for each of the twenty sets of mice.

$$\scriptstyle \begin{array}{c} \hline \text { In Magnetic Field } & \text { Not in Magnetic Field } \\ \hline\end{array}$$

$$\scriptstyle\begin{array}{cccc} \text { Cage } & \text { Weight Gain (g) } & \text { Cage } & \text { Weight Gain (g) } \\ \hline 1 & 22.8 & 11 & 23.5 \\ 2 & 10.2 & 12 & 31.0 \\ 3 & 20.8 & 13 & 19.5 \\ 4 & 27.0 & 14 & 26.2 \\ 5 & 19.2 & 15 & 26.5 \\ 6 & 9.0 & 16 & 25.2 \\ 7 & 14.2 & 17 & 24.5 \\ 8 & 19.8 & 18 & 23.8 \\ 9 & 14.5 & 19 & 27.8 \\ 10 & 14.8 & 20 & 22.0 \\ \hline \end{array}$$

Test whether the variances of the two sets of weight gains are significantly different. Let $\alpha=0.05 .$ For the mice in the magnetic field, $s_{X}=5.67 ;$ for the other mice, $s_{Y}=3.18$.

Particle accelerators, such a s the Large Hadron Collider, use magnetic fields to steer charged particles a round a ring. Consider a proton ring with 36 identical bending mag nets connected by straight segments. The protons move along a 1.0-m-long circular arc as they pass through each magnet. What magnetic field strength is needed i n each magnet to steer protons around the ring with a speed of $2.5 \times 10^{7} \mathrm{m} / \mathrm{s}$? Assume that the field is uniform inside the magnet, zero outside.

Find the relative rate of change of f(x) at the indicated value of x. Round to three decimal places. f(x)=420-5 x; x=25

Suppose an experimenter intends to do a regression analysis by taking a total of 2n data points, where the $x_{i}$'s are restricted to the interval [0,5]. If the xy-relationship is assumed to be linear and if the objective is to estimate the slope with the greatest possible precision, what values should be assigned to the $x_{i}$'s?

Monochromatic light at 577 nm illuminates a diffraction grating with 325 lines/mm. Determine the highest order that can be observed with this grating at the given wavelength.