The temperature of the sun's core is about 1.5 ×10^7 K. Assuming the core to consist of atomic

The temperature of the sun's core is about 1.5 ×10^7 K....

The temperature of the sun's core is about $1.5 \times 10^{7} \mathrm{~K}$. Assuming the core to consist of atomic hydrogen gas and recalling that temperature measures the average kinetic energy of the atoms, compute $(a)$ the thermal energy of $1 \mathrm{~kg}$ of the gas and $(b)$ the mass associated with this energy. $\left[\bar{E}_{k}=3 k T / 2\right],$ where $k$ is the Boltzmann constant (see Chapter 3 ). $]$

Key Concepts

Equipartition Theorem

This principle in statistical mechanics states that energy is equally shared among all available degrees of freedom of a system. For an ideal gas, it implies that each translational degree of freedom contributes equally to the energy of the system, which leads to the formula for average kinetic energy per particle (3/2 kT) in three dimensions.

Boltzmann Constant

The Boltzmann constant (k) is a fundamental physical constant that connects the average kinetic energy of particles in a gas with the temperature. It acts as a bridge between macroscopic and microscopic physics, allowing temperature in Kelvin to be related to energy in joules.

Thermal Energy Calculation

Thermal energy in a gas is determined by summing the kinetic energy of all individual particles. Using the average kinetic energy per particle, one can multiply by the number of particles (found by dividing the total mass by the mass of an individual particle) to determine the total thermal energy contained in a given mass of gas.

Mass-Energy Equivalence

Mass-energy equivalence, expressed by Einstein’s equation E = mc², establishes a relationship between energy and mass. It allows one to calculate the equivalent mass that corresponds to a given amount of energy, highlighting the principle that mass can be converted into energy and vice versa.

Transcript

00:01 So this question belongs to the modern physics in which we have temperature of the sun's core t is equal to 1 .5 multiplied by 10 to the power 7 kelvin and assuming that core is consist of atomic hydrogen gas and recalling the temperature measurement the average kinetic energy of the atom.

00:19 So for the first part we have to calculate the thermal energy okay, e -thermal.

00:24 So we know that the energy associated with the energy associated with the temperature per atom, it is equals to e, k, this can be written as 3x2 kb multiplied by temperature t where kb is the b b b b b b b, so substituting values, so 3 by 2 multiplied by kb which is 1 .38, multiplied by 10 to the power minus 23 jjoules per kelvin and temperature t which is 1 .5 midplied by 10 to the power 7 kelvin.

00:53 So from here we get ek, this is equals to 3 .1 mudplabbit 10 to the power minus 716 jule per atom.

01:01 And we have given mass m equals to 1 kg.

01:04 So in the 1kg there will be some number of atoms and this will be equals to 1 by mass of one atomic hydrogen that is u 1 am u.

01:14 So 1 divided by u which is which is equals to 1 .67 multiplied by 10 to the power minus 27.

01:21 So the total thermal energy associated with 1 kg e thermal this will be equals to e k divided by n.

01:30 Sorry, multiplied by n because these are total numbers of atoms.

01:35 So substituting values, so ek, which is this value.

01:38 So 3 .1 multiplied by 10 to the power minus 16, multiplied by n, which is this complete value.

01:43 So 1 divided by 1 .67 multiplied by 10 to the power minus 27...

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