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**Thermodynamics adiabat - help**

**1. Homework Statement**

Vertical cylindrical bottle is closed by a fixed piston (mass m). Under the piston in volume [itex]V_1[/itex] there is 1 mole of ideal 1-atomic gas with pressure [itex]P_1[/itex]. Over the piston there is vacuum, cross section area is equal to S. The piston becomes free and begins moving. There goes an adiabatic process. Friction between piston and walls of bottle is very small. Calculate gas temperature in the moment of

**first**stop of the piston.

**2. Homework Equations**

We have a system of three equations with three variables.

[tex] p(V_1+Sh)=RT [/tex]

[tex] mgh=c_v(T-T_1)[/tex]

[tex] p_1V_1^\frac {5} {3} = pV^ \frac {5} {3}[/tex]

First is equation of state, second is energy conservation, third - adiabatic process. h - difference in height of piston before moving and in the moment of first stop. [itex]T_1[/itex] is gas temperature in the beginning. [itex]c_v[/itex] - heat capacity in constant volume process (it's equal in this case to [itex]\frac {3} {2}R[/itex]).

**3. The Attempt at a Solution**

After some algebra I got a single equation determining T:

[tex]T^3(aT+b)^2=c[/tex]

where a, b, c - constants equal to:

[tex]a=3RS[/tex]

[tex]b=(2mg-3p_1S)V_1[/tex]

[tex]c=T_1^3V_1^2m^2g^2[/tex]

It seems to me that this equation has one real solution but I can't find it.

Please help me.

Thank you.

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