How much energy would be required to break a helium nucleus into its constituents, two protons and two neutrons? The rest masses of a proton (including an electron), a neutron, and neutral helium are, respectively, $1.00783 \mathrm{u}$, $1.00867 \mathrm{u}$, and $4.00260 \mathrm{u}$. (This energy difference is called the total binding energy of the ${ }_2^4 \mathrm{He}$ nucleus.)

Show that the nonzero elements in $\mathbb { Z } _ { n }$ form a group under the product $[ a ] [ b ] = [ a b ]$ if and only if $n$ is a prime.

You draw a card at random from a standard deck of 52 cards. Find each of the following conditional probabilities: a) The card is a heart, given that it is red. b) The card is red, given that it is a heart. c) The card is an ace, given that it is red. d) The card is a queen, given that it is a face card.

What is a greenhouse gas? Give three examples.

The cesium isotope ${ }^{137} \mathrm{Cs}$, with $Z=55$, is radioactive and decays by beta decay. A beta particle is observed in the laboratory with a kinetic energy of $300 \mathrm{keV}$. The nucleus of a ${ }^{137} \mathrm{Cs}$ atom has a diameter of $12.4 \mathrm{fm}$. With what kinetic energy was the beta particle ejected from the ${ }^{137} \mathrm{Cs}$ nucleus?

Consider again the conditions of the mentioned exercise. Suppose that several parts will be observed and that the different parts are conditionally independent given each of the three states of repair of the machine. If seven parts are observed and exactly one is defective, compute the posterior probabilities of the three states of repair.

Are the alveoli located only in the gas exchange zone?

MCU Phone Company sells its cordless phone for $300 per unit. Fixed costs total$540,000, and variable costs are $120 per unit. Calculate the (1) contribution margin per unit and (2) break-even point in units.

Atmospheric air is normally modeled as an ideal gas.

Write a balanced equation for each of the following nuclear reactions: (a) mercury-180 decays into platinum- 176 (b) zirconium- 90 and an electron are produced by the decay of an unstable nucleus (c) thorium-232 decays and produces an alpha particle and a radium- 228 nucleus, which decays into actinium-228 by beta decay (d) neon-19 decays into fluorine- 19

If a diffraction grating produces its third-order bright band at an angle of $78.4^{\circ}$ for light of wavelength $681 \mathrm{~nm}$, find (c) Will there be a fourth-order bright band? Explain.

How does gas solubility depend on temperature?

If the nucleus is a few fm in diameter, the distance between the centers of two protons must be $\approx 2 \mathrm { fm }$. a. Calculate the repulsive electric force between the centers of two protons that are 2.0 fm apart. b. Calculate the attractive gravitational force between two protons that are 2.0 fm apart. Could gravity be the force that holds the nucleus together?

A gold nucleus contains 197 nucleons (79 protons and 118 neutrons) packed tightly against each other. A single nucleon (proton or neutron) has a radius of about $1 \times 10^{-15}\ \mathrm{m}.$ Remember that the volume of a sphere is $\frac{4}{3} \pi r^{3}.$ Calculate the approximate radius of the gold nucleus.