?Calculate the fractions of molecules in a gas that have a speed in a range \(\Delta v\)

\(\alpha\)-Terpineol is a pleasant-smelling oil obtained from pine. How many allylic hydrogens does it have? Is the hydroxyl group allylic? Text Transcription: alpha

Write a second resonance contributor for each of the following. Is the charge or unpaired electron shared equally by both allylic carbons? If not, which one bears more of the charge or unpaired electron?

The two compounds shown differ by a factor of 60 in their first-order rate constants for hydrolysis in 50% ethanol water at 45°C. Which is more reactive? Why?

What three alcohols are the expected products of the SN1 hydrolysis of the compound shown?

From among the following compounds, choose the two that yield the same carbocation on ionization.

Assume that N-bromosuccinimide serves as a source of \(\mathrm{Br_2}\), and write equations for the propagation steps in the formation of 3-bromocyclohexene by allylic bromination of cyclohexene. Text Transcription: \mathrm{Br_2}

After heating a solution of allyl tert-butyl sulfide and sodium ethoxide in ethanol for several hours, tert-butyl propenyl sulfide was isolated in 66% yield. Suggest a stepwise mechanism for this isomerization. Which has the smaller pKa (is the stronger acid), the reactant or the product?

Many naturally occurring substances contain several carbon–carbon double bonds: some isolated, some conjugated, and some cumulated. Identify the types of carbon–carbon double bonds found in each of the following substances: (a) \(\beta\)-Springene (a scent substance from the dorsal gland of springboks) (b) Cembrene (occurs in pine resin) (c) The sex attractant of the male dried-bean beetle Text Transcription: beta

Dicarbonyl compounds such as quinones are reactive dienophiles. (a) 1,4-Benzoquinone reacts with 2-chloro-1,3-butadiene to give a single product \(\mathrm{C}_{10} \mathrm{H}_{9} \mathrm{ClO}_{2}\) in 95% yield. Write a structural formula for this product. (b) 2-Cyano-1,4-benzoquinone undergoes a Diels–Alder reaction with 1,3-butadiene to give a single product \(\mathrm{C}_{11} \mathrm{H}_{9} \mathrm{NO}_{2}\) in 84% yield. What is its structure? Text Transcription: C_10H_9ClO_2 C_11H_9NO_2

2,3-Di-tert-butyl-1,3-butadiene is extremely unreactive in Diels–Alder reactions. Explain.

Methyl acrylate (H2C=CHCO2CH3) reacts with 1,3-cyclopentadiene to give a mixture of two products. Write structural formulas for both and predict which one predominates.

The compound shown undergoes an intramolecular Diels–Alder reaction at room temperature. What is the structure of the product?

What diene and dienophile could you use to prepare the compound shown?

A Cope rearrangement of a divinylcyclopropane derivative was the final step in a laboratory synthesis of (R)-(–)-dictyopterene C, a constituent of Pacific seaweed. Deduce the structure, including stereochemistry of the reactant.

What is the product of the following reaction?

Give an acceptable IUPAC name for each of the following compounds: (a) \(\mathrm{H}_{2} \mathrm{C}=\mathrm{CH}\left(\mathrm{CH}_{2}\right)_{5} \mathrm{CH}=\mathrm{CH}_{2}\) (c) \(\left(\mathrm{H}_{2} \mathrm{C}=\mathrm{CH}\right)_{3} \mathrm{CH}\) (f) \(\mathrm{H}_{2} \mathrm{C}=\mathrm{C}=\mathrm{CHCH}=\mathrm{CHCH}_{3}\) Text Transcription: H_2C =CH(CH_2)_5CH= CH_2 (H_2C= CH)_3CH H_2C= C= CHCH= CHCH_3

A certain species of grasshopper secretes an allenic substance of molecular formula \(\mathrm{C}_{13} \mathrm{H}_{20} \mathrm{O}_{3}\) that acts as an ant repellent. The carbon skeleton and location of various substituents in this substance are indicated in the partial structure shown. Complete the structure, adding double bonds where appropriate. Text Transcription: C_13 H_20 O_3

(a) Describe the molecular geometry expected for 1,2,3-butatriene (\(\mathrm{H_{2} C=C=C=CH_2}\)). (b) Two stereoisomers are expected for 2,3,4-hexatriene (\(\mathrm{CH_3CH=C=C=CHCH_3}\)). What should be the relationship between these two stereoisomers? Text Transcription: \mathrm{H_{2} C=C=C=CH_2} \mathrm{CH_3CH=C=C=CHCH_3}

The pKa of CH3CH=CHCH=CH2 has been estimated to be about 35. Give the structure of its conjugate base and use resonance to show the sharing of negative charge among the various carbons.

(a) What compound of molecular formula \(\mathrm{C}_{6} \mathrm{H}_{10}\) gives 2,3-dimethylbutane on catalytic hydrogenation over platinum? (b) What two compounds of molecular formula \(\mathrm{C}_{11} \mathrm{H}_{20}\) give 2,2,6,6-tetramethylheptane on catalytic hydrogenation over platinum? Text Transcription: C_6 H_10 C_11 H_20

Give the structure, exclusive of stereochemistry, of the principal organic product formed on reaction of 2,3-dimethyl-1,3-butadiene with each of the following: (a) \(2 \ \mathrm{~mol} \ \mathrm{H}_{2}\), platinum catalyst (b) 1 mol HCl (product of 1,2-addition) (c) 1 mol HCl (product of 1,4-addition) (d) \(1 \ \mathrm{~mol} \ \mathrm{Br}_{2}\) (product of 1,2-addition) (e) \(1 \ \mathrm{~mol} \ \mathrm{Br}_{2}\) (product of 1,4-addition) (f) \(2 \ \mathrm{~mol} \ \mathrm{Br}_{2}\) Text Transcription: 2 mol H_2 1 mol Br_2 2 mol Br_2

Bromination of 1,5-cyclooctadiene with N-bromosuccinimide (NBS) gives a mixture of two constitutional isomers of \(\mathrm{C}_{8} \mathrm{H}_{11} \mathrm{Br}\). Suggest reasonable structures for these two isomers. Text Transcription: C_8 H_11 Br

Identify the more reactive dienophile in each of the following pairs.

Which of the following two dienes can undergo the Diels–Alder reaction? Explain.

Two constitutional isomers of molecular formula \(\mathrm{C}_{\mathrm{8}} \mathrm{H}_{12} \mathrm{O}\) are formed in the following reaction. Ignoring stereochemistry, suggest reasonable structures for these Diels–Alder adducts. Text Transcription: C_8 H_12 O

The two stereoisomers of cinnamic acid (C6H5CH=CHCO2H) each give a different product on reaction with 1,3-butadiene. Write an equation for each reaction.

Acetylene resembles ethylene in that, although being a poor dienophile itself, it adds to dienes when its triple bond bears electron-attracting substituents. The reaction of 1,3-butadiene with diethyl acetylenedicarboxylate gives a Diels–Alder adduct in 98% yield. What is that product?

Heating the compound shown at \(220^{\circ} \mathrm{C}\) gave a mixture of two stereoisomeric Claisen rearrangement products in approximately equal amounts. Write a structural formula for each. Text Transcription: 220^circ C

Specify and analyse critically the assumptions that underlie the kinetic model of gases.

Provide molecular interpretations for the dependencies of the mean free path on the temperature, pressure, and size of gas molecules.

Use the kinetic model of gases to explain why light gases, such as He, are rare in the Earth’s atmosphere but heavier gases, such as O2, CO2, and N2, once formed remain abundant.

Determine the ratios of (i) the mean speeds, (ii) the mean translational kinetic energies of \(\mathrm{H}_{2}\) molecules and Hg atoms at \(20^{\circ} \mathrm{C}\). Text Transcription: H_2 20 degree C

Determine the ratios of (i) the mean speeds, (ii) the mean translational kinetic energies of He atoms and Hg atoms at \(25^{\circ} \mathrm{C}\). Text Transcription: 25 degree C

Calculate the root-mean-square speeds of H2 and O2 molecules at \(20^{\circ} \mathrm{C}\). Text Transcription: 20 degree C

Calculate the root-mean-square speeds of CO2 molecules and He atoms at 20 °C.

Use the Maxwell–Boltzmann distribution of speeds to estimate the fraction of \(\mathrm{N}_{2}\) molecules at 400 K that have speeds in the range \(200-210 \mathrm{m} \mathrm{s}^{-1}\). Hint: The fraction of molecules with speeds in the range v to v + dv is equal to f(v)dv, where f(v) is given by eqn 1B.4. Text Transcription: N_2 200-210 ms^-1

Use the Maxwell–Boltzmann distribution of speeds to estimate the fraction of \(\mathrm{CO}_{2}\) molecules at 400 K that have speeds in the range \(400-405 \mathrm{ms}^{-1}\). See the hint in Exercise E1B.3(a). Text Transcription: CO_2 400-405 ms^-1

What is the relative mean speed of N2 and H2 molecules in a gas at \(25^{\circ} \mathrm{C}\)? Text Transcription: 25 degree C

What is the relative mean speed of O2 and N2 molecules in a gas at \(25^{\circ} \mathrm{C}\)? Text Transcription: 25 degree C

Calculate the most probable speed, the mean speed, and the mean relative speed of \(\mathrm{CO}_{2}\) molecules at \(20^{\circ} \mathrm{C}\). Text Transcription: CO_2 20 degree C

Calculate the most probable speed, the mean speed, and the mean relative speed of \(\mathrm{H}_{2}\) molecules at 20^{\circ} \mathrm{C}\). Text Transcription: H_2 20 degree C

Evaluate the collision frequency of H2 molecules in a gas at 1.00 atm and \(25^{\circ} \mathrm{C}\). Text Transcription: 25 degree C

Evaluate the collision frequency of O2 molecules in a gas at 1.00 atm and \(25^{\circ} \mathrm{C}\). Text Transcription: 25 degree C

Assume that air consists of \(\mathbf{N}_{2}\) molecules with a collision diameter of 395 pm. Calculate (i) the mean speed of the molecules, (ii) the mean free path, (iii) the collision frequency in air at 1.0 atm and \(25^{\circ} \mathrm{C}\). Text Transcription: N_2 25 degree C

The best laboratory vacuum pump can generate a vacuum of about 1 nTorr. At \(25^{\circ} \mathrm{C}\) and assuming that air consists of \(\mathbf{N}_{2}\) molecules with a collision diameter of 395 pm, calculate at this pressure (i) the mean speed of the molecules, (ii) the mean free path, (iii) the collision frequency in the gas. Text Transcription: N_2 25 degree C

At what pressure does the mean free path of argon at \(20^{\circ} \mathrm{C}\) become comparable to the diameter of a \(100 \mathrm{cm}^{3}\) vessel that contains it? Take \(\sigma=0.36 \mathrm{nm}^{2}\). Text Transcription: 20 degree C 100cm^3 Sigma = 0.36nm^2

At what pressure does the mean free path of argon at \(20^{\circ} \mathrm{C}\) become comparable to 10 times the diameters of the atoms themselves? Take \(\sigma=0.36 \mathrm{nm}^{2}\). Text Transcription: 20 degree C Sigma = 0.36nm^2

At an altitude of 20 km the temperature is 217 K and the pressure is 0.050 atm. What is the mean free path of \(\mathbf{N}_{2}\) molecules? \(\sigma=0.43 \mathrm{nm}^{2}\). Text Transcription: sigma = 0.43 nm^2 N_2

At an altitude of 15 km the temperature is 217 K and the pressure is 12.1 kPa. What is the mean free path of \(\mathbf{N}_{2}\) molecules? \(\sigma=0.43 \mathrm{nm}^{2}\). Text Transcription: sigma = 0.43 nm^2 N_2

A rotating slotted-disc apparatus consists of five coaxial 5.0 cm diameter discs separated by 1.0 cm, the radial slots being displaced by \(2.0^{\circ}\) between neighbours. The relative intensities, I, of the detected beam of Kr atoms for two different temperatures and at a series of rotation rates were as follows: Find the distributions of molecular velocities, \(f\left(v_{x}\right)\), at these temperatures, and check that they conform to the theoretical prediction for a one-dimensional system for this low-pressure, collision-free system. Text Transcription: 2.0 degree f(v_x)

Consider molecules that are confined to move in a plane (a two-dimensional gas). Calculate the distribution of speeds and determine the mean speed of the molecules at a temperature T.

A specially constructed velocity-selector accepts a beam of molecules from an oven at a temperature T but blocks the passage of molecules with a speed greater than the mean. What is the mean speed of the emerging beam, relative to the initial value? Treat the system as one-dimensional.

What, according to the Maxwell–Boltzmann distribution, is the proportion of gas molecules having (i) more than, (ii) less than the root mean square speed? (iii) What are the proportions having speeds greater and smaller than the mean speed? Hint: Use mathematical software to evaluate the integrals.

Calculate the fractions of molecules in a gas that have a speed in a range \(\Delta v\) at the speed \(n v_{m p}\) relative to those in the same range at \(v_{m p}\) itself. This calculation can be used to estimate the fraction of very energetic molecules (which is important for reactions). Evaluate the ratio for n = 3 and n = 4. Text Transcription: Deltav nv_mp v_mp

Derive an expression for \(\left\langle v^{n}\right\rangle^{1 / n}\) from the Maxwell–Boltzmann distribution of speeds. Hint: You will need the integrals given in the Resource section, or use mathematical software. Text Transcription: Leftangle V^n rightangle^l/n

Calculate the escape velocity (the minimum initial velocity that will take an object to infinity) from the surface of a planet of radius R. What is the value for (i) the Earth, \(R=6.37 \times 10^{6} \mathrm{m}\), \(g=9.81 \mathrm{m} \mathrm{s}^{-2}\), (ii) Mars, \(R=3.38 \times10^{6} \mathrm{m}\), \(m_{\mathrm{Mars}} / m_{\mathrm{Earth}}=0.108\). At what temperatures do \(\mathrm{H}_{2}\), He, and \(\mathrm{O}_{2}\) molecules have mean speeds equal to their escape speeds? What proportion of the molecules have enough speed to escape when the temperature is (i) 240 K, (ii) 1500 K? Calculations of this kind are very important in considering the composition of planetary atmospheres. Text Transcription: R = 6.37 x 10^6m G = 9.81 m s^-2 R = 3.38 x 10^6m m_mars/m_earth = 0.108 H_2 O_2

Plot different Maxwell–Boltzmann speed distributions by keeping the molar mass constant at \(100 \mathrm{g} \mathrm{mol}^{-1}\) and varying the temperature of the sample between 200 K and 2000 K. Text Transcription: 100 g mol^-1

Evaluate numerically the fraction of \(\mathrm{O}_{2}\) molecules with speeds in the range \(100 \mathrm{ms}^{-1}\) to \(200 \mathrm{ms}^{-1}\) in a gas at 300 K and 1000 K. Text Transcription: O_2 100 ms^-1 200 ms^-1

The maximum in the Maxwell–Boltzmann distribution occurs when df(v)/dv = 0. Find, by differentiation, an expression for the most probable speed of molecules of molar mass M at a temperature T.

methane, \(\mathrm{CH}_{4}\), molecule may be considered as spherical, with a radius of 0.38 nm. How many collisions does a single methane molecule make if 0.10 mol \(\mathrm{CH}_{4}(\mathrm{g})\) is held at \(25^{\circ} \mathrm{C}\) in a vessel of volume \(1.0 \mathrm{dm}^{3}\)? Text Transcription: CH_4(g) 25 degree C 1.0dm^3