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Chapter 39: Problem 45

What is the de Broglie wavelength of a red blood cell, with mass \(1.00 \times 10^{-11} \mathrm{g}\) , that is moving with a speed of 0.400 \(\mathrm{cm} / \mathrm{s}\) ? Do we need to he 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 \(1.6565 \times 10^{-17}\) m, negligible for blood flow analysis.

Step by step solution

01

Convert Mass to Kilograms

First, we must convert the mass of the red blood cell from grams to kilograms. The mass is given as \(1.00 \times 10^{-11}\) g. To convert this to kilograms, use the conversion: 1 g = \(1.00 \times 10^{-3}\) kg. Thus, the mass in kilograms is: \(1.00 \times 10^{-11} \times 10^{-3} = 1.00 \times 10^{-14}\) kg.
02

Apply de Broglie Wavelength Formula

The de Broglie wavelength \( \lambda \) is calculated using the formula \( \lambda = \frac{h}{mv} \), where \( h \) is Planck's constant (\( 6.626 \times 10^{-34} \text{ m}^2 \text{ kg/s} \)), \( m \) is the mass in kg, and \( v \) is the velocity in m/s. First, convert the velocity from cm/s to m/s: \( 0.400 \text{ cm/s} = 0.00400 \text{ m/s} \).
03

Calculate de Broglie Wavelength

Substituting the values into the formula: \( \lambda = \frac{6.626 \times 10^{-34}}{(1.00 \times 10^{-14}\text{ kg})(0.00400\text{ m/s})} \). Simplifying the calculation gives: \( \lambda = \frac{6.626 \times 10^{-34}}{4.00 \times 10^{-17}} = 1.6565 \times 10^{-17} \text{ meters} \).
04

Interpretation of the Wavelength

The wavelength calculated (\(1.6565 \times 10^{-17}\) meters) is extremely small. In practical terms, this wavelength is far too small to have any significant effect on the macroscopic flow of blood. As a result, the wave nature of red blood cells can be ignored when considering 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.

Red Blood Cell
Red blood cells are fascinating components of blood with unique structures and functions. These small, round cells are primarily responsible for transporting oxygen from the lungs to tissues all over the body. They then carry carbon dioxide back to the lungs for exhalation. This vital task is supported by their hemoglobin content, a protein that binds oxygen. Blood cells are typically 6-8 micrometers in diameter, and despite their tiny size, they perform monumental work in keeping us alive. Red blood cells lack a nucleus, which throughout evolutionary history, provided them with more space to pack in hemoglobin and thereby optimize oxygen delivery. In scientific problems like the de Broglie wavelength calculation, we consider their mass. Here, it's crucial to convert this mass from grams to kilograms for precision, especially when dealing with tiny masses, like that of a single red blood cell.
Wave Nature of Particles
In the quantum world, it's important to understand that even small particles can exhibit wave-like properties. This concept, known as the wave-particle duality, was proposed by physicists such as Louis de Broglie. According to de Broglie, particles such as electrons, protons, and even red blood cells can behave like waves. This wave nature is fundamentally characterized by the de Broglie wavelength. The concept of wave nature might seem strange when imagining tangible, larger objects like red blood cells. However, when scales become very small, classical physical laws give way to quantum physics. Yet, the calculated wavelength of red blood cells demonstrates that their wave properties are insignificant on a macroscopic scale, like blood flow, due to their extremely short wavelength.
Mass Conversion
In physics, unit conversion is vital to ensuring the accuracy of calculations, especially with small quantities like the mass of a red blood cell. Such conversions help facilitate the application of physical formulas effectively. To convert the mass, you multiply by the conversion factor, knowing that 1 gram equals 0.001 kilograms. This step is essential to harmonize units such as meters per second used in other calculations. For our problem, converting the mass of a single red blood cell from grams to kilograms is straightforward:
  • Given mass in grams: \(1.00 \times 10^{-11}\)
  • Converted to kilograms: \(1.00 \times 10^{-11} \times 10^{-3} = 1.00 \times 10^{-14}\) kg
This small mass is crucial for the next stages of solving for the de Broglie wavelength.
Velocity Conversion
Just like with mass, converting velocity to the correct units is an important step in our problem-solving process. Velocity is initially given in centimeters per second (cm/s), which is essential to convert into meters per second (m/s) to align with standard SI units in physics formulas. To convert from cm/s to m/s, you divide by 100 (since 1 meter equals 100 centimeters):
  • Given velocity: 0.400 cm/s
  • Converted to meters/sec: \(0.400 \div 100 = 0.00400\) m/s
With this conversion, we ensure consistency across calculations, such as when applied in the de Broglie wavelength formula, where precision is key to deriving meaningful results.

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Most popular questions from this chapter

Why Don't We Diffract? (a) Calculate the de Broglie wavelength of a typical person walking through a doorway. Make reasonable approximations for the necessary quantitics. (b) Will the person in part (a) exhibit wave-like behavior when walking through the "single slit" of a doorway? Why?

In a TV picture tube the accelerating voltage is 15.0 \(\mathrm{kV}\) , and the electron beam passes through an aperture 0.50 \(\mathrm{mm}\) in diameter to a screen 0.300 \(\mathrm{m}\) away. (a) Calculate the uncertainty in the component of the electron's velocity perpendicular to the line between aperture and screen. (b) What is the uncertainty in position of the point where the electrons strike the screen? (c) Does this uncertainty affect the clarity of the picture significantly? (Use nonrelativistic expressions for the motion of the electrons. This is fairly accurate and is certainly adequate for obtaining an estimate of uncertainty effects.)

You want to study a biological spocimen by mcans of a wavelength of \(10.0 \mathrm{nm},\) and you have a choice of using electromagnetic waves or an electron microscope. (a) Calculate the ratio of the energy of a 10.0 -nm- wavelength photon to the kinetic energy of a 10.0 -nm-wavelength electron. (b) In view of your answer to part (a), which would be less damaging to the specimen you are studying; photons or electrons?

A beam of 40 -V electrons traveling in the \(+x\) direction passes through a slit that is parallel to the \(y\) -axis and 5.0\(\mu \mathrm{m}\) wide. The diffraction pattern is recorded on a screen 2.5 \(\mathrm{m}\) from the slit. (a) What is the de Broglie wavelength of the electrons? (b) How much time does it take the electrons to travel from the slit to the screen? (c) Use the width of the central diffraction pattern to calculate the uncertainty in the \(y\) -component of momentum of an electron just after it has passed through the slit. (d) Use the result of part (c) and the Heisenberg uncertainty principle (Eq, 39.11 for \(y\) ) to estimate the minimum uncertainty in the \(y\) -coordinate of an electron just after it has passed through the slit. Compare your result to the width of the slit.

A particle is described by the normalized wave function \(\psi(x, y, z)=A x e^{-\alpha x^{2}} e^{-\beta \beta} e^{-\gamma x^{2}},\) where \(A, \alpha, \beta,\) and \(\gamma\) are all real, positive constants. The probability that the particle will be found in the infinitesimal volume \(d x d y d z\) centered at the point \(\left(x_{0}, y_{0}, z_{0}\right)\) is \(\left|\psi\left(x_{0}, y_{0}, z_{0}\right)\right|^{2} d x d y d z\) (a) At what value of \(x_{0}\) is the particle most likely to be found? (b) Are there values of \(x_{0}\) for which the probability of the particle being found is zero? If so, at what \(x_{0} ?\)
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