Numerical Problems on Cell Structure and Function:

1. Calculating cell size from a micrograph: $$Actual\ Cell\ Length = Magnification\ Factor\ ×\ Measured\ Length $$ $$Actual\ Cell\ Length = 10,000\times (0.05 × 10^{-3}\ mm)$$ $$Actual\ Cell\ Length = 50\ µm$$

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

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

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

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

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

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

8. Estimating the efficiency of cellular respiration: $$Useful\ Work = Efficiency × (Enthalpy\ of\ Combustion) × (Mass\ of\ Glucose)$$ $$Useful\ Work = (0.4) × (6800\ cal/g) × (1\ g)$$ $$Useful\ Work = 2720\ cal$$ $$Useful\ Work = 2.72 kcal$$

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: $$Osmotic\ Pressure = MRT$$ $$Osmotic\ Pressure = (0.1\ M) × (0.082\ L \ atm/ {K \ mol}) × (298\ K)$$ $$Osmotic\ Pressure = 2.48\ atm$$

Numerical Problems on Biomolecules:

1. Converting between molar mass and molecular weight: $$Molecular\ Weight = Number\ of\ Atoms × Atomic\ Mass$$ $$Molecular\ Weight = (12\ × 12\ g/mol) + (22\ × 1\ g/mol) + (11\ × 16\ g/mol)$$ $$Molecular\ Weight = 144\ g/mol + 22\ g/mol + 176\ g/mol$$ $$Molecular Weight = 342\ g/mol$$

2. Calculating the percentage composition of an element in a compound: $$Percentage\ Composition = \frac{Mass\ of\ Element}{Molecular\ Weight} × 100$$ $$Percentage\ Composition\ of\ Carbon = \frac{(6 × 12\ g/mol)}{180\ g/mol}\ × 100$$ $$Percentage\ Composition\ of\ Carbon = 40%$$

3. Determining the empirical formula of a compound: $$Empirical\ Formula = \frac{\text{Mass %}}{ \text{Atomic Mass}} \times \frac{\text{Smallest mole value}}{ \text{Lowest ratio}}$$ $$Empirical\ Formula = \frac{40%}{12\ g/mol} : \frac{6.67%}{1\ g/mol} : \frac{53.33%}{16\ g/mol}$$ $$Empirical\ Formula = \frac{3.33}{12}:\frac{6.67}{1}:\frac{3.33}{16}$$ $$Simplifying\ the\ ratio\ yields\ the\ empirical\ formula: CH_2O$$

4. Estimating the number of moles of solute in a solution: $$Number\ of\ Moles = \frac{Mass\ of\ Solute}{Molecular\ Weight }$$ $$Number of Moles = \frac{10\ g}{100\ g/mol} $$ $$Number of Moles = 0.1\ mol$$

5. Calculating the concentration of a solution: $$Concentration = \frac{Number\ of\ Moles }{ Volume\ of\ Solution }$$ $$Concentration = \frac{0.1\ mol}{1\ L}$$ $$Concentration = 0.1\ M$$

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: $$Turnover\ Number = \frac{Rate\ of\ Reaction}{Enzyme\ Concentration}$$ $$Turnover\ Number = \frac{10^{-5}\ mol/min }{1 \times 10^{-6}\ M}$$ $$Turnover\ Number = 10\ s^{-1}$$

9. Estimating the specificity of an enzyme: $$Specificity\ Constant = \frac{k_{cat}}{K_m}$$ $$Probability\ of\ Reaction = Specificity\ Constant × Concentration\ of\ Substrate $$ $$Probability = (10^3\ M^{-1}s^{-1}) × (1\ M)$$ $$Probability = 1000\ s^{-1}$$ 10. Determining the number of possible primary structures of a protein: $$Number\ of\ Primary\ Structures = n^L$$ $$Number\ of\ Primary\ Structures = 20^{200}$$