Final answer:
The difference in energy between deoxygenated and oxygenated states of hemoglobin at 300K, with a given ratio of 821:1, is 0.40 kcal/mol. This is calculated using the Beer-Lambert law, the gas constant, and the provided temperature.
Step-by-step explanation:
To find the difference in energy (kcal) between deoxygenated and oxygenated states of hemoglobin at 300K, we use the given ratio of deoxygenated to oxygenated hemoglobin (821:1) and the provided Boltzmann equation p1/p2 = exp[(E2-E1)/RT]. In this equation, p1 and p2 represent the probability of the molecules being in state 1 and state 2, respectively. R is the gas constant, T is the temperature in K, and (E2-E1) is the energy difference between the two states in kcal. To solve for (E2-E1), we rearrange the Boltzmann equation to get (E2-E1) = ln(p1/p2) * (RT).
Given that R = 1.987 cal/K/mol, T = 300K, and the ratio of p1/p2 (deoxygenated/oxygenated hemoglobin) is 821, we can plug these values into the equation:
(E2-E1) = ln(821) * (1.987 cal/K/mol * 300K)/1000 = (6.709) * (595.8 cal/mol)/1000 = 3.999 kcal/mol
However, we need to convert calories to kilocalories (1 kcal = 1000 cal), dividing by 1000, which gives us:
(E2-E1) = 0.40 kcal/mol