Book edition 10th Edition

Author(s) David Halliday

Pages 1328 pages

ISBN 9781118230718

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Short Answer

Assume that the protons in a hot ball of protons each have a kinetic energy equal tokT, where k is the Boltzmann constant and T is the absolute temperature. If $T = 1 \times 10^{7} K$ , what (approximately) is the least separation any two protons can have?

The least separation any two protons can have is 1pm.

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Write the given data

Kinetic energy of each proton, K = kT
Temperature of the system, T = $1 \times 10^{7} K$

Step 2: Determine the concept of least separation

For the least separation between any two protons, the energy considered should be the maximum for the given system. Again due to the conservation of energy within the system the total maximum energy is converted to the potential energy of the system at the distance of the least separation. Thus, using this concept, the required separation value is calculated.

Formula:

The potential energy of the two charged system is given as follows:

$U = \frac{q_{1} q_{2}}{4 π ε_{0} r}$ ….. (i)

Step 3: Calculate the least separation between any two protons

As per the given data, the kinetic energy of each proton is given as:

$K = k_{B} T$

Substitute the values and solve as:

$K = (1.38 \times 10^{- 23} \frac{J}{K}) (1 \times 10^{7} K) = 1.38 \times 10^{16} J$

At the closest separation, all the kinetic energy is converted to potential energy. Thus, using the value in equation (i), the total kinetic energy of the two-proton system can be given that is used to calculate the least separation between the two protons as follows:

2K = U

$2 K = \frac{e^{2}}{4 {πε}_{0} r_{\min}} r_{\min} = \frac{e^{2}}{4 {πε}_{0} (2 K)}$

Substitute the values and solve as:

$r_{\min} = \frac{(9 \times 10^{9} \frac{N . m^{2}}{C^{2}}) {(1.6 \times 10^{- 19} C)}^{2}}{2 (1.38 \times 10^{- 16} J)} = 8.33 \times 10^{- 13} m \approx 1 pm$

Hence, the value of the least separation is 1 pm .

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Fundamentals Of Physics

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Short Answer

Assume that the protons in a hot ball of protons each have a kinetic energy equal tokT, where k is the Boltzmann constant and T is the absolute temperature. If $T = 1 \times 10^{7} K$ , what (approximately) is the least separation any two protons can have?

Step by Step Solution

Step 1: Write the given data

Step 2: Determine the concept of least separation

Step 3: Calculate the least separation between any two protons

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Assume that the protons in a hot ball of protons each have a kinetic energy equal tokT, where k is the Boltzmann constant and T is the absolute temperature. If T=1×107K , what (approximately) is the least separation any two protons can have?

Step by Step Solution

Step 1: Write the given data

Step 2: Determine the concept of least separation

Step 3: Calculate the least separation between any two protons

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Assume that the protons in a hot ball of protons each have a kinetic energy equal tokT, where k is the Boltzmann constant and T is the absolute temperature. If $T = 1 \times 10^{7} K$ , what (approximately) is the least separation any two protons can have?