# 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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