Let's solve the problem step by step. ## **Given Data** - Time, \( t = 19. \) s - Droplet volume, \( V = 3.9 \) mL \( = 3.9 \times 10^{-6} \) m³ (since 1 mL = \( 10^{-6} \) m³) - Surface tension, \( \gamma = .0229 \) J/m² - Dynamic viscosity, \( \eta = .091 \) Pa·s - **We are to find:** Spot radius \( R \) after 19. sec in **mm** (4 significant figures) --- ## **Step 1: Identify thereading Law** Since the initial height is smaller than the capillary length, gravity can be neglected, and we use the **Tanner's Law** (for small drops spreading on a surface): \[ R(t) = \left(\frac{A \gamma V^3 t}{\eta}\right)^{1/10} \] Where: - \( R(t) \) = radius at time \( t \) - \( A \) is a dimensionless constant (for a perfectly wetting droplet, \( A \approx .136 \)), often used in literature. --- ## **Step 2: Plug in the Values** Given: - \( V = 3.9 \times 10^{-6} \) m³ - \( t = 19. \) s - \( \gamma = .0229 \) J/m² - \( \eta = .091 \) Pa·s - \( A = .136 \) Plug into the equation: \[ R = \left( \frac{.136 \times .0229 \times (3.9 \times 10^{-6})^3 \times 19.}{.091} \right)^{1/10} \] --- ## **Step 3: Calculate the Numerator** Calculate \( (3.9 \times 10^{-6})^3 \): \[ (3.9 \times 10^{-6})^3 = 3.9^3 \times 10^{-18} = 59.319 \times 10^{-18} \] Now, calculate the numerator: \[ .136 \times .0229 \times 59.319 \times 10^{-18} \times 19. \] First, \( .136 \times .0229 = .0031144 \) Then, \( .0031144 \times 59.319 = .1846 \) Then, \( .1846 \times 19. = 3.507 \) So, total numerator is \( 3.507 \times 10^{-18} \) --- ## **Step 4: Divide by Viscosity** \[ \frac{3.507 \times 10^{-18}}{.091} = 3.853 \times 10^{-17} \] --- ## **Step 5: Take the 1/10th Power** \[ R = (3.853 \times 10^{-17})^{.1} \] Take log to make the calculation easier: \[ \log_{10}(3.853 \times 10^{-17}) = \log_{10}(3.853) + \log_{10}(10^{-17}) = .586 + (-17) = -16.414 \] Dividing by 10: \[ -16.414 / 10 = -1.6414 \] So, \[ 10^{-1.6414} \approx .02274 \] --- ## **Step 6: Convert to mm** \[ R = .02274 \text{ m } = 22.74 \text{ mm} \] --- ## **Final Answer (to 4 significant figures)** \[ \boxed{22.74 \ \text{mm}} \] --- ### **Summary** - Formula used: Tanner's Law for droplet spreading - Plugged in given values step by step - Final radius after 19. s is **22.74 mm** (to 4 significant figures)