Ge108: Homework 1Due: Wednesday, Oct 5. In classSeptember 29, 2005Problem 1: Radiative CoolingThe atmosphere radiates to space in the infrared at a rate σT4powerarea , where σis the Stephan-Boltzmannconstant. The atmosphere absorbs sunlight at a rate F,powerarea , and currently F=240 W/m2. The atmo-spheric heat capacity, C, is 1000 J/(K kg). The mass per unit area is P/g where P is the pressure and g isgravity. P=105Pa and g=9.8 m/s2. Therefore,PgCdTdt= F − σT4(1.1)Find the Equilibrium Temperature, Te=Fσ1/4.Let T=Te+δT, so that T4=(Te+δT)4. Use the Taylor Series expansion to express T4. Specifically,T4= Te4+ δT A + (δT )2B + (δT )3C + .... (1.2)Where A, B, and C can be expressed as a function of y=(Te+x)4and are given byA =dydxx=0B =12!d2ydx2x=0C =13!d3ydx3x=0Find A, B, and CAssume the following:dTedt= 0, butδTdt6= 0 and δT  TeWhy can you neglect the (δT)2and (δT)3terms in equation 1.2? Show that if you neglect these terms,you can get an equation of the formdδTdt= −λδT (1.3)This is an equation for exponential decay. Find 1/λ, which is called the radiative cooling time, radiativerelaxation time, or the radiative time constant. Use the values given above. This is the time it takes for thetemperature of the atmosphere to revert back to equilibrium.1Problem 2: Atmospheric Scale HeightAn fluid is said to be in hydrostatic equilibrium if the internal forces are balanced. Lets suppose our fluid isthe atmosphere with density ρ and we’re looking at a slab with thickness, dz, and area A-see figure 1. Wecan use the force equation F=ma to find the equation for hydrostatic equilibrium. Recall that [P]=hforceareai,so that• PA = downward force due to pressure• ρgAdz = downward force from gravity• (P+dP)A =upward force due to pressureUsing this information, you can then show that balancing the forces gives youdP = −ρgdz, (2.1)which has a minus sign because the dP/dz is negative. This gives us the differential equation that describeshydrostatic equilibrium.dPdz= −ρ(z)g(z) (2.2)For the moment, lets ignore the fact that g changes with height (it changes slowly near the planet’s surfacecompared to other parameters). You can use the Ideal Gas Law (PV=NRT) to relate ρ to P. Specifically,dZ}APP+dPFigure 1: A slab of atmosphere with area A, thickness dZ, and density ρ.2note the following:NV=number of moleculesvolumem =average massmolecule= average molecular massρ =mNV,so that P can now be written asP = ρRTm(2.3)Use eq. 2.3 to eliminate ρ in eq. 2.2 and show that eq 2.2 can be rewritten as followsdPdz= −1HP (2.4)Note that this equation describes an exponential decay. Find H in terms of R, T, m, and g and expressit in terms of km. R=8.314 J/K mole, T=Tefrom Problem 1, m=0.029 kg/mol, g=9.8 m/s2.Problem 3: Thermal EquilibriumWater at a temperature, T0is placed outside where the temperature is Te. The equation for change intemperature isdTdt=1τ(Te− T ), (3.1)where τ is the time constant. Solve for T(t) assuming Teis constantProblem 4Four mineral grains in a rock contain the following isotopes expressed in ppm86Sr87Sr87RbSample 1 29.6 21.1 5.93Sample 2 40.2 29.4 21.7Sample 3 19.7 14.7 15.1Sample 4 33.4 25.5 36.5Plot the Rb-Sr whole rock isocron and find the age of the rock3Problem 5Consider a contaminated reservoir of volume V. Let µ be the the mixing ratio (Think of it as the number ofcontaminant molecules per water molecule). Fresh water enters the reservoir and contaminated water leavesat a rate R (volume per unit time). Show that the mass of the contaminant decays exponentially-specifically show that µ decreases over time as an exponential decay. Find the time constantτ as a function of the parameters