Numerical Problems on Cell Structure and Function:

1. Calculating cell size from a micrograph: $$ActualCellLength=MagnificationFactor\times MeasuredLength$$ $$ActualCellLength=10,000\times (0.05\times {10}^{-3}mm)$$ $$ActualCellLength=50\mu m$$

2. Measuring cell membrane thickness: $$ThicknessofCellMembrane=\frac{\lambda }{4(n_2-n_1)}$$ $$ThicknessofCellMembrane=\frac{550\times {10}^{-9}m}{(4\times (1.5-1.33))}$$ $$ThicknessofCellMembrane\approx 34.4\times {10}^{-9}m$$ $$ThicknessofCellMembrane\approx 34.4nm$$

3. Determining the surface area of a spherical cell: $$SurfaceAreaofaSphere=4\pi r^2$$ $$SurfaceArea=4\pi (5\times {10}^{-6}m{)}^2$$ $$SurfaceArea=4\times 3.14\times (25\times {10}^{-12}{m}^2)$$ $$SurfaceArea=3.14\times 100\times {10}^{-12}{m}^2$$ $$SurfaceArea=314\times {10}^{-10}{m}^2$$ $$SurfaceArea=314\mu m^2$$

4. Estimating the volume of a cuboidal cell: $$Volume=Length\times Width\times Height$$ $$Volume=(10\times {10}^{-6}m)\times (20\times {10}^{-6}m)\times (5\times {10}^{-6}m)$$ $$Volume=1000\times {10}^{-18}{m}^3$$ $$Volume=1000n{m}^3$$

5. Calculating the mass of a cell: $$Mass=Density\times Volume$$ $$Mass=(1g/c{m}^3)\times [(4/3)\pi \times r^3]$$ $$Mass=(1g/c{m}^3)\times [(4/3)\times 3.14\times (5\times {10}^{-4}cm{)}^3]$$ $$Mass\approx 1.05\times {10}^{-12}g$$ $$Mass\approx 1.05pg$$

6. Determining the concentration of a cellular compound: $$NumberofMolecules=Concentration\times Volume$$ $$NumberofMolecules=(0.1/100)\times (1000\times {10}^{-15}L)$$ $$NumberofMolecules={10}^{-17}mol$$ $$NumberofMolecules=0.1molecules$$

7. Calculating the rate of diffusion: $$RateofDiffusion=\frac{Distance}{Time}$$ $$RateofDiffusion=\frac{(10\times {10}^{-6}m)}{(t)}$$ $$t=\frac{(10\times {10}^{-6}m)}{(10\times {10}^{-5}c{m}^2/s)}$$ $$t={10}^{-1}s$$

8. Estimating the efficiency of cellular respiration: $$UsefulWork=Efficiency\times (EnthalpyofCombustion)\times (MassofGlucose)$$ $$UsefulWork=(0.4)\times (6800cal/g)\times (1g)$$ $$UsefulWork=2720cal$$ $$UsefulWork=2.72kcal$$

9. Determining the pH of a cellular compartment: $$pH=-log[H^{+}]$$ $$pH=-log({10}^{-7})$$ $$pH=7$$

10. Calculating the osmotic pressure of a solution: $$OsmoticPressure=MRT$$ $$OsmoticPressure=(0.1M)\times (0.082Latm/Kmol)\times (298K)$$ $$OsmoticPressure=2.48atm$$

Numerical Problems on Biomolecules:

1. Converting between molar mass and molecular weight: $$MolecularWeight=NumberofAtoms\times AtomicMass$$ $$MolecularWeight=(12\times 12g/mol)+(22\times 1g/mol)+(11\times 16g/mol)$$ $$MolecularWeight=144g/mol+22g/mol+176g/mol$$ $$MolecularWeight=342g/mol$$

2. Calculating the percentage composition of an element in a compound: $$PercentageComposition=\frac{MassofElement}{MolecularWeight}\times 100$$ $$PercentageCompositionofCarbon=\frac{(6\times 12g/mol)}{180g/mol}\times 100$$ $$PercentageCompositionofCarbon=40$$

3. Determining the empirical formula of a compound: $$EmpiricalFormula=\frac{\text{Mass \%}}{\text{Atomic Mass}}\times \frac{\text{Smallest mole value}}{\text{Lowest ratio}}$$ $$EmpiricalFormula=\frac{40}{12g/mol}:\frac{6.67}{1g/mol}:\frac{53.33}{16g/mol}$$ $$EmpiricalFormula=\frac{3.33}{12}:\frac{6.67}{1}:\frac{3.33}{16}$$ $$Simplifyingtheratioyieldstheempiricalformula:C{H}_{2}O$$

4. Estimating the number of moles of solute in a solution: $$NumberofMoles=\frac{MassofSolute}{MolecularWeight}$$ $$NumberofMoles=\frac{10g}{100g/mol}$$ $$NumberofMoles=0.1mol$$

5. Calculating the concentration of a solution: $$Concentration=\frac{NumberofMoles}{VolumeofSolution}$$ $$Concentration=\frac{0.1mol}{1L}$$ $$Concentration=0.1M$$

6. Determining the pH of a solution: $$pH=-log[H^{+}]$$ $$pH=-\log ({10}^{-4})$$ $$pH=4$$

7. Estimating the pKa of a weak acid: $$pKa=-log(K_a)$$ $$pKa=-\log ({10}^{-5})$$ $$pKa=5$$

8. Calculating the rate of an enzyme-catalyzed reaction: $$TurnoverNumber=\frac{RateofReaction}{EnzymeConcentration}$$ $$TurnoverNumber=\frac{{10}^{-5}mol/min}{1\times {10}^{-6}M}$$ $$TurnoverNumber=10s^{-1}$$

9. Estimating the specificity of an enzyme: $$SpecificityConstant=\frac{k_{cat}}{K_m}$$ $$ProbabilityofReaction=SpecificityConstant\times ConcentrationofSubstrate$$ $$Probability=({10}^3{M}^{-1}{s}^{-1})\times (1M)$$ $$Probability=1000s^{-1}$$ 10. Determining the number of possible primary structures of a protein: $$NumberofPrimaryStructures=n^L$$ $$NumberofPrimaryStructures={20}^{200}$$