Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Interdisciplinary paper IDP1 / 2

Take a fixed amount of dry air, with no phase change, moving quasistatically so that its pressure matches the surrounding atmosphere. From the first law of thermodynamics and $H=U+pV$,For adiabatic motion $dq=0$. Treating the parcel as an ideal gas of fixed composition makes its enthalpy depend only on temperature, so for $n$ moles $dH=n{C}_{p}dT$ even though the pressure changes. Hence $n{C}_{p}dT=Vdp$. The environment is in hydrostatic equilibrium, $dp/dz=-\rho g$, and the parcel is taken to have the ambient density at the displacement point. Using $\rho =nM/V$ givesHere $C_p$ is molar and $c_p=C_p/M$ is mass-specific heat capacity. Positive $z$ points upward; the commonly quoted positive dry adiabatic lapse rate is ${\Gamma }_d=-dT/dz$. Approximate constancy of $g$ and heat capacity gives a constant gradient. Condensation, external heating, entrainment or composition changes invalidate this dry-parcel relation.

Convert the molar mass to $0.0288\mathrm{kg}\mathrm{mo}{\mathrm{l}}^{-1}$ before substitution:The often-used environmental value near $-6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ is less steep because rising moist air can condense water vapor, releasing latent heat and offsetting some expansion cooling. A saturated parcel therefore follows a smaller-magnitude moist adiabatic lapse rate, which varies with temperature, pressure and water content. Real environmental profiles also reflect radiation, mixing and weather systems; $6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ is a representative profile value, not a universal parcel law.

Much stratospheric heating is supplied by absorption of solar ultraviolet radiation by oxygen and ozone, whereas carbon dioxide is an important emitter of thermal infrared radiation. In the usual stratospheric balance, increasing carbon dioxide enhances net infrared cooling at the original temperature without supplying a matching increase in the solar heating. Temperature must fall until infrared emission and absorption again balance the heating.

A useful optically thin schematic is $Q_{\mathrm{solar}}=c_{\mathrm{C}{\mathrm{O}}_2}K[B(T)-\overline{J}]$, where $B(T)$ represents the relevant thermal source function and $\overline{J}$ the incident infrared radiation field. With positive fixed solar heating and an approximately fixed radiation field, increasing $c_{\mathrm{C}{\mathrm{O}}_2}$ lowers the required $B(T)-\overline{J}$, hence lowers $T$. This explains why the surface greenhouse argument cannot simply be applied locally to the stratosphere. Spectral blocking of upwelling radiation contributes too; the real response depends on altitude, ozone heating and radiative transfer, rather than emissivity alone.^[ref]

Put $X=[{\mathrm{O}}_3]$ and $u=T^{-1}$. The fast oxygen–ozone partition obeysDividing the production–loss balance by two and substituting givesUnder the specified restriction, all temperature dependence lies in the three rate coefficients: hold the photodissociation rates and the concentrations of ${\mathrm{O}}_2$, $\mathrm{M}$ and NO fixed. Then $d\ln A/du=-2100-510=-2610\mathrm{K}$ and $d\ln B/du=-1200\mathrm{K}$. Defining $Y=d\ln X/du$, differentiation yieldsThereforeThe factor two in the denominator multiplying $L_C$ comes from the additional dependence $[\mathrm{O}]\propto X/k_4$; ignoring that dependence would give the wrong sensitivity. This derivative has units of kelvin.

Using the stated loss ratio $L_N=3L_C$ in the derived logarithmic sensitivity givesThe positive sign means that cooling, which increases $T^{-1}$, increases the steady ozone concentration within this restricted middle-stratosphere model.

The change in inverse temperature is $\Delta u=1/290-1/300=1.14943\times 10^{-4}{\mathrm{K}}^{-1}$. Treating the sensitivity as approximately constant over the interval,Thus the predicted concentration increases by about 15.3%. Linearizing the exponential gives a rougher 14.3% estimate; a logarithmic change is not itself the exact fractional change.

An independent finite-change calculation checks this approximation without keeping the loss ratio artificially fixed. Normalize the initial Chapman and nitrogen-cycle loss terms to 1 and 3, so the initial production is 4. If $r=X_{290}/X_{300}$, the new steady balance is $e^{-2610\Delta u}{r}^2+3e^{-1200\Delta u}r=4$. Its positive root is $r=1.15342$, essentially the same result.

Lower winter temperatures favour formation and persistence of polar stratospheric clouds, providing particle surfaces on which reservoir chlorine can be activated, for exampleThese are heterogeneous surface processes; cooling changes particle abundance, adsorption, phase and reaction uptake, so a simple gas-phase Arrhenius argument is insufficient. Removal of nitric acid in sedimenting particles can also delay conversion of active chlorine back into chlorine nitrate. The relevant condensation thresholds depend on pressure and water/nitric-acid abundances, rather than a single universal temperature.

Once sunlight is present, $\mathrm{C}{\mathrm{l}}_2+\mathrm{h}\nu \to 2Cl$ provides active chlorine. In the chlorine monoxide dimer cycle,The net reaction is $2{\mathrm{O}}_3\to 3{\mathrm{O}}_2$, with chlorine regenerated. Low temperature favours formation of the ClOOCl dimer and suppresses its thermal dissociation, making photodissociation rather than thermal return of two ClO radicals more competitive. Laboratory flow experiments formed the dimer at 220–240 K and distinguished its infrared and ultraviolet signatures, supporting the chemical identity needed for this mechanism.^[ref]

Colder polar conditions can prepare and sustain greater springtime ozone loss, but darkness alone does not drive the photolytic cycle. In polar night reactive precursors can accumulate; sunlight is required for rapid dimer-cycle ozone destruction. This differs from the middle-stratospheric gas-phase cooling response derived above.^[ref]