Question

A hypodermic syringe is attached to a needle that has an internal radius of \(0.300 \mathrm{mm}\) and a length of \(3.00 \mathrm{cm} .\) The needle is filled with a solution of viscosity $2.00 \times 10^{-3} \mathrm{Pa} \cdot \mathrm{s} ;\( it is injected into a vein at a gauge pressure of \)16.0 \mathrm{mm}$ Hg. Ignore the extra pressure required to accelerate the fluid from the syringe into the entrance of the needle. (a) What must the pressure of the fluid in the syringe be in order to inject the solution at a rate of $0.250 \mathrm{mL} / \mathrm{s} ?$ (b) What force must be applied to the plunger, which has an area of \(1.00 \mathrm{cm}^{2} ?\)

Short answer

Answer: To find the force applied to the plunger, follow these steps: 1. Convert the flow rate to m^3/s: 0.250 mL/s * 10^(-6) m^3/mL = 0.250 * 10^(-6) m^3/s. 2. Convert the gauge pressure to Pascals: (16.0 mm Hg / 760 mm Hg) * 101325 Pa. 3. Use Poiseuille's law to find the pressure difference: ΔP = (8 * η * L * Q) / (π * r^4). 4. Calculate the pressure in the syringe: P_syringe = P_gauge + ΔP. 5. Calculate the force on the plunger: F = P_syringe * A.

Step-by-step solution

(Step 1: Find the flow rate)

We are given the flow rate is \(0.250 \mathrm{mL} / \mathrm{s}\), which we need to convert to \(\mathrm{m}^3 / \mathrm{s}\) by using the relationship \(\mathrm{mL} = 10^{-6}\mathrm{m}^3\): $$Q=0.250 \times 10^{-6} \,\mathrm{m}^{3}/\mathrm{s}$$

(Step 2: Convert pressure to SI units)

The gauge pressure is given as \(16.0 \mathrm{mm}\) Hg. We need to convert it to Pascals (Pa) using the standard atmospheric pressure (\(101325 \mathrm{Pa}\)) and the relationship between mm Hg and atmospheric pressure, which is \(1 \,\mathrm{atm} = 760 \,\mathrm{mm} \, \mathrm{Hg}\): $$P_{gauge} = \frac{16.0}{760}\times 101325 \,\mathrm{Pa}$$

(Step 3: Apply Poiseuille's law to find pressure difference)

Recall that the Poiseuille's law states, $$Q = \frac{\Delta P \pi r^4}{8\eta L}$$ Given the flow rate \(Q\), viscosity \(\eta\), internal radius \(r\) and length \(L\) of the needle, we can solve for the pressure difference \(\Delta P\): $$\Delta P = \frac{8\eta LQ}{\pi r^4}$$

(Step 4: Calculate pressure in the syringe)

Since the pressure difference \(\Delta P\) is equal to the pressure in the syringe minus the gauge pressure \(P_{gauge}\), we can find the pressure in the syringe: $$P_{syringe} = P_{gauge} + \Delta P$$

(Step 5: Calculate the force on the plunger)

We know that the pressure \(P_{syringe}\) is equal to the force on the plunger divided by the area of the plunger \(A\). Hence, we can derive the force on the plunger by multiplying the pressure in the syringe with the area of the plunger: $$F = P_{syringe} \times A$$