Gas in a piston

Helium gas is enclosed in a gas-tight piston which is held in position by the clamping force of an elastic spring. In mechanical equilibrium, the gas pressure p=FAp = -\frac FA corresponds to the tensioning force FF of the spring divided by the area AA. The gas is then brought to a temperature of T=600KT = 600 \,\text{K} by rapid heating from room temperature T=300KT = 300 \,\text{K}.

What is the final temperature of the gas (in units of Kelvin) after setting a new mechanical balance with the spring?

Details and Assumptions:

  • The volume of the gas is V=Ax,V = Ax, where AA is the cross-sectional area and xx is the piston position.
  • The mechanical force of the spring equals F=kx,F = - k x, where kk is the spring constant.
  • The piston does not move during the heating from 300K300 \,\text{K} to 600K600 \,\text{K}.
  • Helium can be treated as an ideal gas.
  • Energy is transferred only by mechanical work. The piston is perfectly thermally insulated.

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