Background Freezing isn't the only challenge in cryopreservation - thawing can be just as difficult. A microwave oven seems like a nice solution, since it deposits energy quickly and microwaves are non-ionizing radiation (they do not cause DNA mutation). However, water absorbs microwaves more effectively than ice does, meaning that the portion of an organ that has already melted will get warmer at a higher rate than the remaining ice – the opposite of what we want! - The transmission of radiation through a weakly absorbing material such as ice or water can be modeled by Beer's law, which assumes that the rate of absorption at a depth x is proportional to the local radiation intensity I(x) times an absorption coefficient, which is often written as μ or a or just µ). Noting that absorption decreases the intensity, we can write a differential equation a Solving the differential equation with the boundary condition on the surface being gives the relationship For a standard microwave oven (f= 2.45 GHz, λ = 12 cm), we can assume that the absorption coefficient µ = 0.2 cm³. The absorption coefficient of ice is lower and highly dependent on temperature. For this problem, let's assume that μice = 0.05 cm³. For this problem, let's thaw a frozen liver for implantation. Liver is over 75% water and one of the more homogeneous organs, so we will model it using standard water properties. Assume that this liver is broad enough that we can analyze it with 1-D rectangular coordinates. We will compare the situation just before the liver starts to melt, to the situation when the top centimeter is thawed but the rest is frozen. The answers will be ratios so you may leave the surface intensity I(0) as an unknown constant. a. What is the ratio of the energy absorbed in the top centimeter when it is frozen to the energy absorbed when it is thawed? b. What is the ratio of the energy absorbed in the next centimeter (which is ice in both cases) when the top centimeter is water to when the top centimeter is ice? c. Suppose that a cup of water in your microwave absorbs five times as much power per mL as a block of ice does. Note that this is power per volume, so the exact size of the cup and ice don't matter here. Do you expect that the water will heat up five times as fast as the ice? Explain your answer, taking material properties into consideration. d. The absorption coefficient of ice just below the melting temperature (range of -5° C to 0° C) is dramatically higher than the coefficient for ice below about -5° C. Therefore, it is beneficial to raise the temperature of a frozen organ to near the melting point by immersion in water before freezing. Would you expect the organ to heat up faster (higher dT/dt) immersed in water while the organ is frozen solid, while it is partially thawed, or while it is completely thawed? Explain your answer.