Gliese 581c is the first Earth-like extrasolar terrestrial planet discovered. Its parent star, Gliese 581, is a red dwarf that radiates electromagnetic waves with power $5.00 \times 10^{24}$W, which is only 1.30% of the power of the Sun. Assume the emissivity of the planet is equal for infrared and for visible light and the planet has a uniform surface temperature. If an average temperature of 287 K is necessary for life to exist on Gliese 581c, what should the radius of the planet’s orbit be?

The equilibrium separation between the atoms of a CO molecule is $0.113 \mathrm{~nm}$. For a molecule, such as $\mathrm{CO}$, that has a permanent electric dipole moment, radiative transitions obeying the selection rule $\Delta \ell=\pm 1$ between two rotational energy levels of the same vibrational level are allowed. (That is, the selection rule $\Delta v=\pm 1$ does not hold.) Find the wavelength of the photons emitted during each transition in (b). In what region of the electromagnetic spectrum are those photons?

The ${ }^{19} \mathrm{~F}$ NMR spectrum of the octahedral ion $\left[\mathrm{PF}_5 \mathrm{Me}\right]^{-}$shows two signals $(\delta-45.8$ and $-57.6$ ppm). Why are two signals observed? From these signals, three coupling constants can be measured: $J_{\mathrm{PF}}=829 \mathrm{~Hz}, J_{\mathrm{FF}}=680 \mathrm{~Hz}$ and $J_{\mathrm{FF}}=$ $35 \mathrm{~Hz}$. Explain the origins of these coupling constants.

There are four isomers with the molecular formula $\mathrm{C}_{4} \mathrm{H}_{9} \mathrm{Cl}$. Only one of these isomers (compound A) has a ­chiral center. When compound A is treated with sodium ethoxide, three products are formed: ­compounds B, C, and D. Compounds B and C are diastereomers, with compound B being the less stable diastereomer. Do you expect compound D to exhibit a signal at approximately 1650 $\mathrm{cm}^{-1}$ in its IR spectrum? Explain.

A device called a Fabry-Perot interferometer contains two parallel mirrors as shown in Figure 4. Multiple reflections occur between the inner mirror surfaces, and the waves transmitted at the top can interfere. Assume that the light waves travel nearly perpendicular to the mirrors, the transmitted waves interfere constructively, and the spacing between the mirrors is exactly $3.5 \mu \mathrm{m}$. What is a possible value for the wavelength in the visible range between $600 \mathrm{~nm}$ and $700 \mathrm{~nm}$ ?

When the 1H NMR spectrum of an alcohol is run in dimethyl sulfoxide (DMSO) solvent rather than in chloroform, exchange of the O-H proton is slow and spin–spin splitting is seen between the O-H proton and C-H protons on the adjacent carbon. What spin multiplicities would you expect for the hydroxyl protons in the following alcohols? (a) 2-Methyl-2-propanol (b) Cyclohexanol (c) Ethanol (d) 2-Propanol (e) Cholesterol (f) 1-Methylcyclohexanol

The adjoining illustration shows four Petri plates used for the Ames test. A piece of filter paper (white circle in the center of each plate) was soaked in one of four preparations and then placed on a Petri plate. The four preparations contained (A) purified water (control), (B) a known mutagen, (C) a chemical whose mutagenicity is under investigation, and (D) the same chemical after treatment with liver homogenate. The number of revertants, visible as colonies on the Petri plates, was determined in each case.

This problem is about how strongly matter is coupled to radiation, the subject with which quantum mechanics began. For a simple model, consider a solid iron sphere 2.00 cm in radius. Assume its temperature is always uniform throughout its volume. Assume Wien's law describes the sphere. Although it emits a spectrum of waves having all different wavelengths, assume its power output is carried by photons of wavelength $\lambda_{\max }$. Find the energy of one photon.

A. A diffraction grating with $5.000 \times 10^{3}$ lines/cm is used to examine the sodium spectrum. Calculate the angular separation of the two closely spaced yellow lines of sodium (588.995 nm and 589.592 nm) in each of the first two orders. Given: Unknown: $5.000 \times 10^{3} \text { lines } / \mathrm{cm}=? \text { lines } / \mathrm{m} \quad \theta=\sin ^{-1}\left(\frac{\mathrm{m} \lambda}{d}\right)$

A miniature spectrometer used for chemical analysis has a diffraction grating with 800 slits/mm set 25.0 mm in front of the detector "screen". The detector can barely distinguish two bright lines that are $30 \mu \mathrm{m}$ apart in the first-order spectrum. What is the resolution of the spectrometer at a wavelength of 600 nm? That is, if two distinct wavelengths can barely be distinguished, one of them being 600.0 nm, what is the wavelength difference $\Delta \lambda$ between the two?

Determine whether the following arguments are best interpreted as being inductive or deductive. Also state the criteria you use in reaching your decision (i.e., the presence of indicator words, the nature of the inferential link between premises and conclusion, or the character or form of argumentation). Each element, such as hydrogen and iron, has a set of gaps—wavelengths that it absorbs rather than radiates. So if those wavelengths are missing from the spectrum, you know that that element is present in the star you are observing.

Predict what you expect to observe in the EPR spectrum of a species in which an unpaired electron interacts with one ${ }^{14} \mathrm{~N}$ nucleus $(I=1)$ and one ${ }^1 \mathrm{H}$ nucleus $(I=b)$ if the hyperfine coupling constants are (a) $A\left({ }^{14} \mathrm{~N}\right)=A\left({ }^1 \mathrm{H}\right)=30 \mathrm{G}$; (b) $A\left({ }^{14} \mathrm{~N}\right)=30 \mathrm{G}$, $A\left({ }^1 \mathrm{H}\right)=10 \mathrm{G}$.

CCOc1cccc(C=NC2CCCCC2N=Cc2cccc(OCC)c2O)c1O The ligand, $\mathrm{H}_2 \mathrm{~L}$, reacts with copper(II) acetate to give a neutral complex. In the ESI mass spectrum of a $\mathrm{CH}_2 \mathrm{Cl}_2 /$ MeOH solution of the product, the base peak is at $\mathrm{m} / \mathrm{z}$ $494.1$ and exhibits an isotope pattem characteristic of one $\mathrm{Cu}$ atom. Assign the peak and, assuming no fragmentarion occurs, suggest a formula for the complex.

Which of the following possible structures for X can be excluded on the basis of its IR spectrum:

$$(a) \mathrm { CH } _ { 3 } \mathrm { COOCH } _ { 2 } \mathrm { CH } _ { 3 }; \text { (b) HOCH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { CHO }; (c) \mathrm { CH } _ { 3 } \mathrm { CH } _ { 2 } \mathrm { COOCH } _ { 3 }; \mathrm { (d) } \mathrm { CH } _ { 3 } \mathrm { CH } _ { 2 } \mathrm { CH } _ { 2 } \mathrm { COOH? }$$