One cubic centimeter of water has a mass of 1.00 x 10-3 kg. (a) Determine the mass of 1.00 m3 of water. (b) Biological substances are 98% water. Assume that they have the same density as water to estimate the masses of a cell that has a diameter of 1.0%m, a human kidney, and a fly. Model the kidney as a sphere with a radius of 4.0 cm and the fly as a cylinder 4.0 mm long and 2.0 mm in diameter
The Correct Answer and Explanation is :
(a) Mass of 1.00 m³ of Water
Given that 1 cubic centimeter (1 cm³) of water has a mass of 1.00 × 10⁻³ kg, we can determine the mass of 1 cubic meter (1 m³) of water as follows:
- Volume Conversion: 1 m³ equals 1,000,000 cm³ (since 1 m = 100 cm, thus 1 m³ = 100³ cm³).
- Mass Calculation: If 1 cm³ of water has a mass of 1.00 × 10⁻³ kg, then 1,000,000 cm³ will have a mass of: [ 1.00 \times 10^{-3} \, \text{kg/cm}^3 \times 1,000,000 \, \text{cm}^3 = 1,000 \, \text{kg} ] Therefore, the mass of 1.00 m³ of water is 1,000 kg.
(b) Estimating Masses of Biological Substances
Assuming biological substances are 98% water and have the same density as water, we can estimate the masses of the given entities.
- Cell with a Diameter of 1.0 µm
- Volume Calculation: Modeling the cell as a sphere:
- Radius (r): 0.5 µm = 0.5 × 10⁻⁶ m
- Volume (V): [ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (0.5 \times 10^{-6} \, \text{m})^3 = 5.24 \times 10^{-19} \, \text{m}^3 ]
- Mass Calculation: Density of water = 1,000 kg/m³.
- Mass: [ \text{Mass} = \text{Density} \times \text{Volume} = 1,000 \, \text{kg/m}^3 \times 5.24 \times 10^{-19} \, \text{m}^3 = 5.24 \times 10^{-16} \, \text{kg} ]
- Adjusting for 98% water content: [ 5.24 \times 10^{-16} \, \text{kg} \times 0.98 = 5.13 \times 10^{-16} \, \text{kg} ] Thus, the mass of the cell is approximately 5.13 × 10⁻¹⁶ kg.
- Human Kidney
- Volume Calculation: Modeling the kidney as a sphere:
- Radius (r): 4.0 cm = 0.04 m
- Volume (V): [ V = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (0.04 \, \text{m})^3 = 2.68 \times 10^{-4} \, \text{m}^3 ]
- Mass Calculation: Density of water = 1,000 kg/m³.
- Mass: [ \text{Mass} = \text{Density} \times \text{Volume} = 1,000 \, \text{kg/m}^3 \times 2.68 \times 10^{-4} \, \text{m}^3 = 0.268 \, \text{kg} ]
- Adjusting for 98% water content: [ 0.268 \, \text{kg} \times 0.98 = 0.263 \, \text{kg} ] Therefore, the mass of the kidney is approximately 0.263 kg.
- Fly
- Volume Calculation: Modeling the fly as a cylinder:
- Diameter: 2.0 mm = 0.002 m, thus radius (r) = 0.001 m
- Length (h): 4.0 mm = 0.004 m
- Volume (V): [ V = \pi r^2 h = \pi (0.001 \, \text{m})^2 \times 0.004 \, \text{m} = 1.26 \times 10^{-8} \, \text{m}^3 ]
- Mass Calculation: Density of water = 1,000 kg/m³.
- Mass: [ \text{Mass} = \text{Density} \times \text{Volume} = 1,000 \, \text{kg/m}^3 \times 1.26 \times 10^{-8} \, \text{m}^3 = 1.26 \times 10^{-5} \, \text{kg} ]
- Adjusting for 98% water content: [ 1.26 \times 10^{-5} \, \text{kg} \times 0.98 = 1.23 \times 10^{-5} \, \text{kg} ] Thus, the mass of the fly is approximately 1.23 × 10⁻⁵ kg.
Explanation:
These calculations are based on the principle that mass equals density multiplied by volume (Mass = Density × Volume). By modeling each biological entity with simple geometric shapes (sphere for the cell and kidney, cylinder for the fly), we can estimate their volumes using standard geometric formulas.