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 \(p = -\frac FA\) corresponds to the tensioning force \(F\) of the spring divided by the area \(A\). The gas is then brought to a temperature of \(T = 600 \,\text{K}\) by rapid heating from room temperature \(T = 300 \,\text{K}\).

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 \(V = Ax,\) where \(A\) is the cross-sectional area and \(x\) is the piston position.
  • The mechanical force of the spring equals \(F = - k x,\) where \(k\) is the spring constant.
  • The piston does not move during the heating from \(300 \,\text{K}\) to \(600 \,\text{K}\).
  • 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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