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Chapter 9: Problem 22

An IV is connected to a patient's vein. The blood pressure in the vein has a gauge pressure of \(12 \mathrm{mm}\) Hg. At least how far above the vein must the IV bag be hung in order for fluid to flow into the vein? Assume the fluid in the IV has the same density as blood.

Short Answer

Expert verified
Answer: The IV bag must be hung at a minimum height of 0.151 meters (or 15.1 cm) above the vein for the fluid to flow into the patient's vein.

Step by step solution

01

Identify the given information

The gauge pressure in the patient's vein is 12 mm Hg. We also know that the density of blood is approximately the same as the fluid in the IV.
02

Convert the pressure

To make the calculation easier, let's convert the gauge pressure from mm Hg to Pascals (Pa). We can use the conversion: 1 mm Hg = 133.3 Pa. Therefore, the gauge pressure in the vein is: \(12 \mathrm{mm}\,\mathrm{Hg} \times 133.3\,\mathrm{Pa/mm}\,\mathrm{Hg} = 1599.6 \, \mathrm{Pa}\)
03

Apply Pascal's Law and the concept of pressure in a fluid column

According to Pascal's law, the pressure applied to an enclosed fluid is transmitted undiminished to all portions of the fluid. Therefore, we can say: \(P_{fluid} = P_{vein}\). The pressure in a fluid column is given by the hydrostatic pressure equation: \(P = h\rho g\), where \(h\) is the height of the liquid column, \(\rho\) is the density, and \(g\) is the acceleration due to gravity.
04

Solve for height h

Since \(P_{fluid} = P_{vein}\), we can write the equation as: \(h\rho g = P_{vein}\). Now we have to solve for the height (h) at which the IV bag must be hung: \(h = \dfrac{P_{vein}}{\rho g}\) The density of blood (and IV fluid) is 1060 kg/m³. The acceleration due to gravity is approximately 9.81 m/s².
05

Calculate the height

Now we can use the values we obtained to calculate the minimum distance above the vein: \(h = \dfrac{1599.6\,\mathrm{Pa}}{ 1060\,\mathrm{kg/m^{3}} \cdot 9.81\,\mathrm{m/s^2}} \approx 0.151\,\mathrm{m}\) So, the IV bag must be hung at least 0.151 meters (or 15.1 cm) above the vein for fluid to flow into the vein.

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