Blood plasma (at 37.0°C) is to be supplied to a patient at the rate of 2.80 × 10−6 m3/s. If the tube connecting the plasma to the patient’s vein has a radius of 2.00 mm and a length of 54.0 cm, what is the pressure difference between the plasma and the patient’s vein? Viscosity of blood plasma is 1.30 × 10−3 Pa·s.

To solve for the pressure difference required to supply blood plasma through a tube at a certain flow rate, we can use **Poiseuille’s Law** for laminar flow, which is:

\[
\Delta P = \frac{8 \eta Q L}{\pi r^4}
\]

where:
- \(\Delta P\) = pressure difference (what we want to find),
- \(\eta\) = viscosity of the fluid (given as \(1.30 \times 10^{-3} \, \text{Pa} \cdot \text{s}\)),
- \(Q\) = volumetric flow rate (given as \(2.80 \times 10^{-6} \, \text{m}^3/\text{s}\)),
- \(L\) = length of the tube (given as 54.0 cm, or 0.54 m),
- \(r\) = radius of the tube (given as 2.00 mm, or 0.002 m).

### Step 1: Substitute the values into Poiseuille’s equation

\[
\Delta P = \frac{8 \cdot (1.30 \times 10^{-3}) \cdot (2.80 \times 10^{-6}) \cdot 0.54}{\pi \cdot (0.002)^4}
\]

### Step 2: Calculate each term

#### 1. Calculate \(r^4\)
\[
r^4 = (0.002)^4 = 1.6 \times 10^{-11} \, \text{m}^4
\]

#### 2. Calculate the numerator
\[
8 \cdot (1.30 \times 10^{-3}) \cdot (2.80 \times 10^{-6}) \cdot 0.54 = 1.57056 \times 10^{-8}
\]

#### 3. Substitute and solve for \(\Delta P\)

\[
\Delta P = \frac{1.57056 \times 10^{-8}}{\pi \cdot 1.6 \times 10^{-11}} 
\]

\[
\Delta P = \frac{1.57056 \times 10^{-8}}{5.0265 \times 10^{-11}}
\]

\[
\Delta P \approx 312.4 \, \text{Pa}
\]

### Answer
The pressure difference required to supply blood plasma at the given rate is approximately:

\[
\Delta P \approx 312.4 \, \text{Pa}
\]

Would you like more details on any of these steps?

---

1. How would changes in the viscosity affect the pressure difference?
2. How would altering the radius impact the flow rate or pressure difference?
3. What assumptions are made when using Poiseuille's Law for blood plasma flow?
4. How would temperature variations affect blood plasma viscosity?
5. Could this calculation apply to other fluids besides blood plasma?

**Tip:** Always ensure units are consistent in physics equations; small discrepancies can lead to significant errors in results.

Blood plasma (at 37.0°C) is to be supplied to a patient at the rate of 2.80 × 10−6 m3/s. If the tube connecting the plasma to the patient’s vein has a radius of 2.00 mm and a length of 54.0 cm, what is the pressure difference between the plasma and the patient’s vein? Viscosity of blood plasma is 1.30 × 10−3 Pa·s.

Find the pressure difference needed to supply blood plasma at a rate of 2.80 × 10⁻⁶ m³/s through a tube with a 2.00 mm radius and 54.0 cm length. This physics problem explores fluid dynamics concepts and applies Poiseuille's Law. Suitable for advanced high school students in Grades 11-12.

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