Blood takes about 1.50 s to pass through a 2.0-mm-long capillary. If the radius of the capillary is 5.00 u�m and the pressure drop is 2.60 kPa, calculate the viscosity of blood
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To calculate the viscosity of blood, we can use Poiseuille's Law, which relates the flow rate of a fluid through a cylindrical tube to its viscosity, pressure drop, and dimensions.
Poiseuille's Law equation: Q = (π * ΔP * r^4) / (8 * η * L)
Where:
Q = Flow rate of blood
ΔP = Pressure drop across the capillary
r = Radius of the capillary
η = Viscosity of blood
L = Length of the capillary
Given:
Pressure drop (ΔP) = 2.60 kPa
Radius (r) = 5.00 μm = 5.00 * 10^(-6) m
Length (L) = 2.0 mm = 2.0 * 10^(-3) m
We need to convert the pressure drop from kilopascals (kPa) to pascals (Pa) before calculating:
ΔP = 2.60 kPa = 2.60 * 10^3 Pa
Now we can substitute the values into the equation and solve for viscosity (η):
Q = (π * ΔP * r^4) / (8 * η * L)
Rearranging the equation to solve for viscosity (η):
η = (π * ΔP * r^4) / (8 * Q * L)
Substituting the given values:
η = (π * (2.60 * 10^3 Pa) * (5.00 * 10^(-6) m)^4) / (8 * Q * (2.0 * 10^(-3) m))
We are given the time it takes blood to pass through the capillary, but we need to convert it to flow rate (Q). Flow rate is the volume of blood passing through per unit time.
Flow rate (Q) = Volume / Time
The volume of blood passing through the capillary can be calculated using the area of the capillary (A) and the length (L):
Volume = A * L
The area of the capillary (A) can be calculated using the formula for the area of a circle:
A = π * r^2
Substituting the given values:
Volume = (π * (5.00 * 10^(-6) m)^2) * (2.0 * 10^(-3) m)
Now we can substitute the values into the equation for flow rate (Q):
Q = Volume / Time
Given:
Time = 1.50 s
Substituting the values:
Q = [(π * (5.00 * 10^(-6) m)^2) * (2.0 * 10^(-3) m)] / (1.50 s)
Now we have the value for flow rate (Q), we can substitute it into our equation for viscosity:
η = (π * (2.60 * 10^3 Pa) * (5.00 * 10^(-6) m)^4) / (8 * [(π * (5.00 * 10^(-6) m)^2) * (2.0 * 10^(-3) m)] / (1.50 s) * (2.0 * 10^(-3) m))
Now we can simplify and calculate the viscosity by canceling out common terms:
η = [(2.60 * 10^3) * (5.00 * 10^(-6))^4 * (1.50)] / [8 * (5.00 * 10^(-6))^2]
Evaluating this equation will give us the viscosity of blood.