On a single plot, qualitatively sketch the distribution of molecular speeds for Kr( g ) at

$$- 50 ^ { \circ } \mathrm { C }$$

Prove that if $\varphi : G \rightarrow G ^ { \prime }$ is a homomorphism, then $\varphi ( G ) ,$ the image of $G ,$ is a subgroup of $G ^ { \prime } .$

True or False. A firm’s profits have little to do with that firm’s ability to grow.

When it comes to making decisions, which of the following statements are true? Select all that apply.

a. Individuals will make the same choices no matter how information is presented.

Hydrogen has two naturally occurring isotopes,

$$^1 H$$

and

$$^2H.$$

Chlorine also has two naturally occurring isotopes,

$$^{35}CI$$

and

$$^{37}Cl$$

Thus, hydrogen chloride gas consists of four distinct types of molecules:

$$^ { 1 } \mathrm { H } ^ { 35 } \mathrm { Cl } , ^ { 1 } \mathrm { H } ^ { 37 } \mathrm { Cl } , ^ { 2 } \mathrm { H } ^ { 35 } \mathrm { Cl } ,$$

and

$$^ { 2 } \mathrm { H } ^ { 37 } \mathrm { Cl }$$

Place these four molecules in order of increasing rate of effusion.

When it comes to making decisions, which of the following statements are true? Select all that apply.

d. Anchoring can help individuals make better decisions.

When it comes to making decisions, which of the following statements are true? Select all that apply.

b. Anchoring is always helpful when making decisions.

Calculate the rms speed for $CO$ molecules at $25^{\circ} C$. What is the ratio of this speed to that of Ar atoms at the same temperature?

Propose a structure consistent with each set of data. a. Compound A: Molecular formula:

$$\mathrm { C } _ { 8 } \mathrm { H } _ { 10 } \mathrm { O }$$

IR absorption at

$$3150 - 2850\ \mathrm { cm } ^ { - 1 }$$

$$^1\text {H NMR data: }$$

1.4 (triplet, 3 H), 3.95 (quartet, 2 H), and 6.8–7.3 (multiplet, 5 H) ppm b. Compound B: Molecular formula:

$$\mathrm { C } _ { 9 } \mathrm { H } _ { 10 } \mathrm { O } _ { 2 }$$

IR absorption at

$$1669\ \mathrm { cm } ^ { - 1 }$$

$$^1\text {H NMR data: }$$

2.5 (singlet, 3 H), 3.8 (singlet, 3 H), 6.9 (doublet, 2 H), and 7.9 (doublet, 2 H) ppm

Factor the trinomial.

$$6 x ^ { 2 } - 14 x - 12$$

Calculate the rms speeds of CO and

$$Cl_2$$

molecules at 300 K.

An infinitely long cylindrical tube, of radius a, moves at constant speed v along its axis. It carries a net charge per unit lenght $\lambda$, uniformly distributed over its surface. Surrounding it, at radius b, is another cylinder, moving with the same velocity but carrying the opposite charge ($-\lambda$). Find: (a) The energy per unit lenght stored in the fields. (b) The momentum per unit lenght in the fields. (c) The energy per unit time transported by the fields across a plane perpendicular to the cylinders.

Place the following gases in order of increasing average molecular speed at

$$25 ^ { \circ } \mathrm { C } : \mathrm { Ne } , \mathrm { HBr } , \mathrm { SO } _ { 2 } , \mathrm { NF } _ { 3 } , \mathrm { CO }$$

Suppose you have two 1-L flasks, one containing

$$N_2$$

at STP, the other containing

$$CH_4$$

at STP. How do these systems compare with respect to number of molecules.