Q71E - College Physics (Urone)

Question

Question: The temperature inside a supernova explosion is said to be 2.00×10¹³ K. (a) What would the average velocity v_rms of hydrogen atoms be? (b) What is unreasonable about this velocity? (c) Which premise or assumption is responsible?

Step-by-Step Solution

Answer

Answer

1. The v_rms of hydrogen atoms are $7.08\times {10}^{8}m/s$.
2. The result is unreasonable because it is a higher value than the speed of light.
3. This unreasonable velocity is due to the given high temperature.

Step 1: Concept

The average velocity of an atom is directly related to its temperature. This is expressed as,

${\mathbf{v}}_{\mathbf{rms}}\mathbf{=}\sqrt{\frac{\mathbf{3}\mathbf{kT}}{\mathbf{m}}}$

Here, is the average velocity, is the Boltzmann constant and its value is . is the temperature in Kelvin and its value is given as 2.00×10¹³ K. is the mass of the atom, and the mass of hydrogen is1.67×10^-27 Kg.

Step 2: Solution for part

Substitute these values into the equation.

$$\begin{gathered} v_{rms}=\sqrt{\frac{3\times (1.38\times {10}^{-23} J/K)\times (2.00\times {10}^{13}K)}{(1.67\times {10}^{-27}Kg)}} \\ =7.04\times {10}^{8}m/s \end{gathered}$$

So, the average velocity of a hydrogen atom is $7.04\times {10}^{8}m/s$  

Step 2: Solution for part (b)

The calculated average velocity is unreasonable because it is faster than the light. The speed of light is approximately equal to . No, objects can exceed speed. So, it is not a possible velocity. 

Step 3: Solution for part (c)

The given premise leads to an unreasonable answer. Here, the given temperature is too high. Since the temperature is directly proportional to the velocity, it gives a higher value.