The density of seawater is 1030 kg/m3 and the density of whole blood is 1050 kg/m3. Note that 1cm3 = 1mL. What is the volume (in Liters, L) of 103 grams seawater? What is the volume (in Liters, L) of 105 grams whole blood?
The correct answer and explanation is:
Let’s solve the problem step by step.
Given:
- Density of seawater, ρsw=1030 kg/m3\rho_{sw} = 1030 \, \text{kg/m}^3
- Density of whole blood, ρwb=1050 kg/m3\rho_{wb} = 1050 \, \text{kg/m}^3
- Mass of seawater, msw=103 gm_{sw} = 103 \, \text{g}
- Mass of whole blood, mwb=105 gm_{wb} = 105 \, \text{g}
- 1 cm3=1 mL1 \, \text{cm}^3 = 1 \, \text{mL}
- 1000 mL=1 L1000 \, \text{mL} = 1 \, \text{L}
Step 1: Convert mass from grams to kilograms
Since density is in kg/m³, convert mass to kg: msw=103 g=1031000=0.103 kgm_{sw} = 103 \, \text{g} = \frac{103}{1000} = 0.103 \, \text{kg} mwb=105 g=1051000=0.105 kgm_{wb} = 105 \, \text{g} = \frac{105}{1000} = 0.105 \, \text{kg}
Step 2: Use the density formula to find volume
Density formula: ρ=mV ⟹ V=mρ\rho = \frac{m}{V} \implies V = \frac{m}{\rho}
- For seawater:
Vsw=0.103 kg1030 kg/m3=1.0×10−4 m3V_{sw} = \frac{0.103 \, \text{kg}}{1030 \, \text{kg/m}^3} = 1.0 \times 10^{-4} \, \text{m}^3
- For whole blood:
Vwb=0.105 kg1050 kg/m3=1.0×10−4 m3V_{wb} = \frac{0.105 \, \text{kg}}{1050 \, \text{kg/m}^3} = 1.0 \times 10^{-4} \, \text{m}^3
Step 3: Convert volume from cubic meters to liters
1 m3=1000 L1 \, \text{m}^3 = 1000 \, \text{L}
- For seawater:
Vsw=1.0×10−4×1000=0.1 LV_{sw} = 1.0 \times 10^{-4} \times 1000 = 0.1 \, \text{L}
- For whole blood:
Vwb=1.0×10−4×1000=0.1 LV_{wb} = 1.0 \times 10^{-4} \times 1000 = 0.1 \, \text{L}
Final answers:
- Volume of 103 g seawater = 0.1 L
- Volume of 105 g whole blood = 0.1 L
Explanation (300 words):
The volume of a substance can be calculated using its mass and density through the relation: V=mρV = \frac{m}{\rho}
where VV is volume, mm is mass, and ρ\rho is density.
In this problem, the densities of seawater and whole blood are given in kilograms per cubic meter (kg/m³), while the mass is given in grams. To maintain unit consistency, the mass must first be converted into kilograms because 1 kg = 1000 g.
Once the mass is converted, we substitute the values into the density formula. For example, seawater with a mass of 103 g converts to 0.103 kg. Using the density of seawater 1030 kg/m31030 \, \text{kg/m}^3, dividing mass by density gives the volume in cubic meters. The calculation yields 1.0×10−4 m31.0 \times 10^{-4} \, \text{m}^3.
Similarly, whole blood with a mass of 105 g converts to 0.105 kg. Using the density 1050 kg/m31050 \, \text{kg/m}^3, the volume is again approximately 1.0×10−4 m31.0 \times 10^{-4} \, \text{m}^3.
Because volumes in cubic meters are often cumbersome for small quantities, converting them to liters is practical, knowing that 1 m3=1000 L1 \, \text{m}^3 = 1000 \, \text{L}. This conversion results in volumes of 0.1 liters for both seawater and whole blood in the given masses.
Interestingly, even though the masses and densities differ slightly, the resulting volumes are nearly the same due to the close values of density and mass, both approximating 0.1 liters.
Understanding these conversions is crucial in many scientific and engineering applications, such as calculating fluid displacement, medical dosage, or chemical solution preparation, where accurate volume measurements are vital.