Expressions for the Maxwell speed distribution for molecules...

Expressions for the Maxwell speed distribution for molecules in a gas are given in Chapter 19. (a) Show that the most probable energy is given by $$ K_{p}=\frac{1}{2} k T$$ Verify this result with the energy distribution curve of Fig. $43-10$ , for which $T=1.5 \times 10^{7} \mathrm{K}$ (b) Show that the most probable speed is given by $$v_{p}=\sqrt{\frac{2 k T}{m}}$$ Find its value for protons at $T=1.5 \times 10^{7} \mathrm{K}$ . (c) Show that the energy corresponding to the most probable speed (which is not the same as the most probable energy is $$ K_{v, p}=k T$$ Locate this quantity on the curve of Fig. $43-10$

Expressions for the Maxwell speed distribution for molecules in a gas are given in Chapter 19. (a) Show that the most probable energy is given by $$
K_{p}=\frac{1}{2} k T$$
Verify this result with the energy distribution curve of Fig. $43-10$ , for
which $T=1.5 \times 10^{7} \mathrm{K}$ (b) Show that the most probable speed is
given by
$$v_{p}=\sqrt{\frac{2 k T}{m}}$$
Find its value for protons at $T=1.5 \times 10^{7} \mathrm{K}$ . (c) Show that the
energy corresponding to the most probable speed (which is not the
same as the most probable energy is $$
K_{v, p}=k T$$
Locate this quantity on the curve of Fig. $43-10$

Key Concepts

Maxwell Speed Distribution

This concept describes the statistical distribution of speeds for particles in an ideal gas. It is derived using kinetic theory and statistical mechanics, predicting how many particles have a given speed at thermal equilibrium. The distribution provides insights into the behavior of gases, including the probabilities of particles having various kinetic energies and speeds based on temperature.

Most Probable Energy

The most probable energy is the value of kinetic energy where the energy probability distribution reaches its maximum. It represents the energy that the largest number of particles possess under the Maxwell distribution. This concept is key in understanding how energy is statistically allocated among particles in a gas and is related to temperature and the Boltzmann constant.

Most Probable Speed

The most probable speed is the speed at which the Maxwell speed distribution attains its maximum value. It indicates the speed that the greatest number of gas molecules are likely to have at a given temperature. Determining the most probable speed helps in linking the microscopic motion of individual particles with macroscopic thermodynamic properties.

Relationship Between Kinetic Energy and Speed

Kinetic energy, given by one half of the mass times the square of the speed, connects the particle's motion to its energy. Due to the quadratic relationship between speed and energy, the peak of the kinetic energy distribution does not coincide with the kinetic energy corresponding to the most probable speed. Understanding this relationship clarifies the subtle distinctions between different statistical measures in the distribution.

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