Consider the Universe to be an adiabatic expansion of atomic hydrogen gas. (a) U

Question

Consider the Universe to be an adiabatic expansion of atomic hydrogen gas. (a) Use the ideal gas law and Equation 12.8a (PV^y=constant) to show that TV^y-1=C, where C is a constant. (b) The current Universe extends at least 15 billion light-years in all directions (1.4 x 10^26 m), and the current temperature of the Universe is 2.7 K. Estimate the temperature of the Universe when it was the size of a nutshell, with a radius of 2 cm. (For this calculation, assume the Universe is spherical.) answer is1.3*10^56K please show work

Explanation / Answer

a)

PV^ = constant

and,

PV = nRT

P = constant * T / V

substituting :

constant * (T / V ) * V^ = constant

TV^-1 = constant

b)

Atomic Hydrogen gas : = 1.4

T1V1^0.4 = T2V2^0.4

Since V R^3

T1 = T2 * (V2/V1)^0.4 = 2.7 * (1.4*10^26 / 0.02)^3*0.4

T1 = 2.7 * (0.7 * 10^28)^1.2

= 7 * 10^33 K

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