Final answer:
To estimate the time for an H2O molecule to undergo a root-mean-square displacement of 1.0 cm with a diffusion coefficient of 2.26 × 10⁻¹ m² s⁻¹, use the formula t = (Xrms²) / (2D). By substituting in the values and calculating, the estimated time can be found.
Step-by-step explanation:
The question asks to estimate the time required for an H2O molecule to undergo a root-mean-square displacement of 1.0 cm (or 0.01 m), given the diffusion coefficient of H2O in water at 25 °C is 2.26 × 10⁻¹ m² s⁻¹. The root-mean-square displacement (Xrms) for diffusion is calculated using the formula Xrms = √(2Dt), where D is the diffusion coefficient and t is the time.
To find the time (t), we rearrange the equation to t = (Xrms²) / (2D). Plugging in the values Xrms = 0.01 m and D = 2.26 × 10⁻¹ m² s⁻¹, we can calculate the time required for this displacement. Performing the calculation gives us t = (0.01²) / (2 × 2.26 × 10⁻¹) = 0.01 / (4.52 × 10⁻¹) s. The final step is to compute the actual numerical value, which gives us the estimated time for an H2O molecule to diffuse 1.0 cm in water at 25 °C.