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Find the temperature of totally ionized hydrogen plasma of density rho = 0.10g//cm^(3) at which the thermal radiation pressure is equal to the gas kinetic pressure of the particles of plasma. Take into account that the thermal radiation pressure p = u//3, where u is the space density of radiation enegry, and at high temperatures all substances obey the equation of state of an ideal gas. |
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Answer» Solution :For an ideal gas `p = nkT` where `n=` number density of the particles and `k = (R )/(N_(A))` is Boltzman constant. In a fully ionized hydrogen plasma, both `H` ions (protons) and electrons contribute to pressure but since the mass of electron is quite small `(=m_(P)//1836)`, only protons contribute to mass density. THUS `n = (2RHO)/(m_(H))` and `p = (2 rhoR)/(N_(A)m_(H))T` where `m_(H) = m_(p)` is the proton or hydrogen mass. equating this to thermal radiation pressure `(2rhoR)/(N_(A)m_(H)) = (u)/(3) = (M_(E))/(3) xx (4)/(3) = (4sigmaT^(4))/(3c)` Then `T^(3) = (3c rhoR)/(2sigmaN_(A)m_(H)) = (3crho R)/(sigma M)` where `M = 2N_(A)m_(H)=` molecular weight of hydrogen `= 2 xx 10^(-3) kg`. Thus `T = ((3c sigmaR)/(sigmaM))^(1//3) = 1.89 xx 10^(7)K` |
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