- #1

CAF123

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## Homework Statement

An interstellar cloud, made up of an ideal gas, collapses with its radius decreasing as $$R = 10^{13} \left(\frac{-t}{216}\right)^{2/3} \text{m}$$ with ##t## measured in years. The time ##t## is taken to be zero at zero radius so that ##t## is always negative.

The cloud collapses isothermally at 10K until its radius reaches 10

^{13}m. It then becomes opaque so that from then on, the collapse takes place adiabatically and reversibly. How many years does it take for the temperature to rise by 800K measured from the time the cloud reaches a radius of 10

^{13}m.

## Homework Equations

The one in question, First Law of Thermodynamics, Adiabatic expansion

## The Attempt at a Solution

The question has not specified how they define 'radius', but I assumed a spherical cloud. Considering the end of the isothermal phase, start of the adiabatic phase, and the end of the adiabatic phase, the following holds: $$\frac{T_i}{P_i^{1-1/\gamma}} = \frac{T_f}{P_f^{1-1/\gamma}},$$ Reexpressing gives: $$T_i^{1/\gamma} V_i^{1-1/\gamma} = T_f^{1/\gamma} V_f^{1-1/\gamma},$$ where ##T_i = 10, V_i = 4/3 \pi (10^{13})^3, T_f = 810K, V_f = 4/3 \pi R^3## When I solve for t, I obtain the incorrect answer.

Perhaps my assumption of the spherical shape of the cloud is incorrect?