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What is the de Broglie wavelength of a red blood cell, with mass 1.00 \(\times\) 10\(^{-11}\) g, that is moving with a speed of 0.400 cm/s? Do we need to be concerned with the wave nature of the blood cells when we describe the flow of blood in the body?

Short Answer

Expert verified
The de Broglie wavelength is \(1.66 \times 10^{-17}\) m; it's too small to affect blood flow.

Step by step solution

01

Convert Units

First, we need to convert the mass of the red blood cell from grams to kilograms and the speed from centimeters per second to meters per second, as these are the standard units used in physics. Mass: \[1.00 \times 10^{-11} \text{ g} = 1.00 \times 10^{-14} \text{ kg}\]Speed: \[0.400 \text{ cm/s} = 0.004 \text{ m/s}\]
02

Use de Broglie Wavelength Formula

The de Broglie wavelength \(\lambda\) is given by the formula:\[\lambda = \frac{h}{mv}\]where:- \(h\) is the Planck constant \(6.626 \times 10^{-34}\) J·s- \(m\) is the mass in kilograms- \(v\) is the velocity in meters per second.
03

Calculate de Broglie Wavelength

Substitute the values into the de Broglie wavelength formula:\[\lambda = \frac{6.626 \times 10^{-34} \text{ J·s}}{1.00 \times 10^{-14} \text{ kg} \times 0.004 \text{ m/s}}\]Calculate:\[\lambda = \frac{6.626 \times 10^{-34}}{4.00 \times 10^{-17}} = 1.66 \times 10^{-17} \text{ m}\]
04

Interpret the Result

The calculated de Broglie wavelength is \(1.66 \times 10^{-17}\) m, which is extremely small compared to the size of a red blood cell, which is typically around \(6 \times 10^{-6}\) m. This means the wave nature of the red blood cell is negligible in the context of describing blood flow in the body.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Wave Nature
The concept of wave nature in particles, including red blood cells, stems from the de Broglie hypothesis. It proposes that all matter exhibits wave-like properties, not just light. This means that everything, from tiny electrons to larger objects like blood cells, has an associated wavelength, known as the de Broglie wavelength. However, this wave nature becomes significant primarily in microscopic particles.

For everyday, macroscopic objects, the wavelength is so tiny that it's practically nonexistent in comparison to the object's size. Therefore, when considering red blood cells or similar biological particles, their wave nature doesn't notably impact their behavior or flow. Since red blood cells in the bloodstream are quite large compared to their minuscule de Broglie wavelength, the wave nature can typically be ignored when examining blood flow.

Understanding this helps explain why quantum mechanics rarely needs to be applied to biological systems at the level of red blood cells. Their physical size and the de Broglie wavelength difference make their quantum effects negligible in the biological processes they are part of.
Planck Constant
The Planck constant is a fundamental constant in physics that plays a crucial role in the study of quantum mechanics. Its value is approximately \(6.626 \times 10^{-34}\) Joule-seconds. This constant is incredibly tiny, making it a bridge between the particle and wave nature of matter. In the de Broglie wavelength formula, the Planck constant facilitates transforming momentum into wavelength, giving insight into how properties typically associated with waves, like wavelength, can apply to particles.

When calculating the de Broglie wavelength of a red blood cell, the Planck constant helps determine whether its wave nature matters in practical scenarios, such as in the flow of blood. Because the constant is so small, objects with larger mass and slower speeds result in nearly infinitesimal wavelengths, which explains why even though matter has wave properties, they are generally not observed in larger objects.

The Planck constant is one of the keystones of quantum physics, highlighting how classical mechanics gives way to quantum mechanics at microscopic scales. It serves as a reminder that while the quantum effects are fascinating, they often remain imperceptible in the everyday world of larger bodies, like human cells.
Blood Flow
Blood flow within the human body is a complicated and critical process, governed by principles of fluid dynamics rather than quantum mechanics. In the context of larger biological structures, such as organs and blood vessels, classical physics provides an accurate and straightforward description. Red blood cells are carried along in the bloodstream due to pressures and forces within the circulatory system which ensures oxygen and nutrients are efficiently distributed throughout the body.

The wave nature of individual red blood cells, while interesting from a theoretical perspective, does not have a noticeable effect on the way blood moves. When calculated, the de Broglie wavelength of such cells turns out to be incredibly small compared to their physical size, confirming that their behavior in the bloodstream is more accurately described by classical physics principles.

Therefore, while quantum mechanics offers profound insights into the nature of matter at microscopic levels, it plays little role in macroscopic processes like blood flow. The movement of blood is mostly determined by factors like heart rate, blood pressure, and the geometry of blood vessels, rendering the quantum wave nature irrelevant in practical medical diagnostics and treatments.

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Most popular questions from this chapter

The brightest star in the sky is Sirius, the Dog Star. It is actually a binary system of two stars, the smaller one (Sirius B) being a white dwarf. Spectral analysis of Sirius B indicates that its surface temperature is 24,000 K and that it radiates energy at a total rate of 1.0 \(\times\) 10\(^{25}\) W. Assume that it behaves like an ideal blackbody. (a) What is the total radiated intensity of Sirius B? (b) What is the peak-intensity wavelength? Is this wavelength visible to humans? (c) What is the radius of Sirius B? Express your answer in kilometers and as a fraction of our sun's radius. (d) Which star radiates more \(total\) energy per second, the hot Sirius B or the (relatively) cool sun with a surface temperature of 5800 K? To find out, calculate the ratio of the total power radiated by our sun to the power radiated by Sirius B.

(a) A particle with mass \(m\) has kinetic energy equal to three times its rest energy. What is the de Broglie wavelength of this particle? (\(Hint\): You must use the relativistic expressions for momentum and kinetic energy: \(E^2 = (pc^2) + (mc^2)^2\) and \(K = E - mc^2\).) (b) Determine the numerical value of the kinetic energy (in MeV) and the wavelength (in meters) if the particle in part (a) is (i) an electron and (ii) a proton.

The negative muon has a charge equal to that of an electron but a mass that is 207 times as great. Consider a hydrogenlike atom consisting of a proton and a muon. (a) What is the reduced mass of the atom? (b) What is the ground-level energy (in electron volts)? (c) What is the wavelength of the radiation emitted in the transition from the \(n\) = 2 level to the \(n\) = 1 level?

Using a mixture of CO\(_2\), N\(_2\), and sometimes He, CO\(_2\) lasers emit a wavelength of 10.6 \(\mu\)m. At power outputs of 0.100 kW, such lasers are used for surgery. How many photons per second does a CO\(_2\) laser deliver to the tissue during its use in an operation?

(a) Using the Bohr model, calculate the speed of the electron in a hydrogen atom in the \(n\) = 1, 2, and 3 levels. (b) Calculate the orbital period in each of these levels. (c) The average lifetime of the first excited level of a hydrogen atom is 1.0 \(\times\) 10\(^{-8}\) s. In the Bohr model, how many orbits does an electron in the \(n\) = 2 level complete before returning to the ground level?

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