Problem 62 - College Physics

Question

Whole blood has a surface tension of \(0.058 \mathrm{~N} / \mathrm{m}\) and a density of \(1050 \mathrm{~kg} / \mathrm{m}^{3}\). To what height can whole blood rise in a capillary blood vessel that has a radius of \(2.0 \times 10^{-6} \mathrm{~m}\) if the contact angle is zero?

Step-by-Step Solution

Answer

The height to which whole blood can rise in a capillary blood vessel that has a radius of \(2.0 \times 10^{-6} \mathrm{~m}\) is approximately \(0.553 \mathrm{~m}\).

Step 1: Identify the given values

Here are the given values: \n\nSurface tension = \(0.058 \mathrm{~N} / \mathrm{m}\) \nDensity of the blood = \(1050 \mathrm{~kg} / \mathrm{m}^{3}\) \nRadius of the capillary = \(2.0 \times 10^{-6} \mathrm{~m}\) \nThe contact angle is zero.

Step 2: Write the capillary rise equation

The formula for the height to which a liquid will rise in a capillary is given by\n\n\[ h = \frac {2s \cos{\theta}}{rg\rho} \]\n\nwhere \n\n\(h\) = height \n\(s\) = surface tension \n\(\theta\) = contact angle \n\(r\) = radius of the capillary \n\(g\) = acceleration due to gravity \n\(\rho\) = density of the liquid\n\nSince the contact angle is zero, and cos(0)=1, this simplifies our equation to \n\n\[ h = \frac {2s }{r \cdot g \cdot \rho} \]

Step 3: Substitute the given values

Now we can substitute the given values into the equation. We also use \(9.81 \mathrm{~m/s^{2}}\) for \(g\) (acceleration due to gravity).\n\nSo we have \n\n\[ h = \frac {2 \cdot 0.058 }{2.0 \times 10^{-6} \cdot 1050 \cdot 9.81} \]

Step 4: Calculate the height

After performing the calculation we find\n\n\[ h \approx 0.553 \mathrm{~m} \]\n\nThis tells us that the blood can rise to a height of approximately 0.553 meters in the capillary.

Key Concepts

Surface Tension

Surface tension might seem like a fairly straightforward concept, but it's a pivotal force in fluid mechanics and plays a critical role in our daily lives and in the workings of our body. Think of surface tension as the 'skin' of a liquid, a cohesive force that binds the liquid's surface molecules together. This tension arises because molecules at the surface of a liquid are not surrounded by similar molecules on all sides, leading them to cohere more tightly with their neighbors on the surface.

In physiology, surface tension is vitally important during processes such as the respiratory function of the lungs, where it helps to keep the alveoli (tiny air sacs in the lungs) from collapsing. A reduction in the surface tension of body fluids like blood can affect the capillary action, which is crucial for the movement of blood through the narrow vessels of the body. Applying this to our textbook problem, the surface tension value for blood, at 0.058 N/m, influences how high it can travel up a capillary blood vessel.

Fluid Dynamics in Physiology

The circulation of blood throughout the body is a key example of fluid dynamics in action within physiology. Blood flows through a complex network of vessels, driven primarily by the pumping action of the heart and influenced by various factors, including blood pressure, vessel elasticity, and indeed, capillary action. Capillary action is the ability of a liquid to flow in narrow spaces without the assistance of, and in opposition to, external forces like gravity.

Understanding the fundamental principles of fluid dynamics helps us comprehend how blood can ascend in the smallest blood vessels, defying gravity's downward pull. This phenomenon has profound implications for ensuring that vital oxygen and nutrients reach every part of the body. Anomalies in these dynamics, like changes in surface tension or blood viscosity, might hint at health issues, making the comprehension of these principles vital for medical professionals.

Capillary Rise Equation

In explaining how fluids move up through narrow tubes, the capillary rise equation is key. It derives from principles in fluid dynamics and is mathematically represented as \[ h = \frac {2s \cos{\theta}}{r g \rho} \], where 'h' stands for the height the liquid can rise, 's' is the surface tension, '\(\theta\)' is the contact angle of the liquid with the wall, 'r' designates the radius of the tube, 'g' is the gravitational acceleration, and '\(\rho\)' is the liquid's density.

The equation shows us that the rise is directly proportional to the surface tension and inversely proportional to both the liquid's density and the radius of the tube. When applying this equation to our textbook example, the blood rises to a height of 0.553 meters in the capillary because of the precise balance between the blood's surface tension and density, the capillary's dimensions, and gravity. In essence, this equation provides a quantitative method to predict fluid behavior in capillaries, which is invaluable for both scientific research and medical diagnostics.