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**1. Homework Statement**

A liquid is enclosed in a metal container that is provided with a piston of the same metal. The system is originally at a pressure of 1.00 atm (1.013*10^5 Pa) and at a temperature of 30.0 C. The piston is forced down until the pressure on the liquid is increased by 50.0 atm, and then clamped in this position.

Find the new temperature at which the pressure of the liquid is again 1.00 atm.

Assume that the cyclider is sufficiently strong so that its volume is not altered by the pressure, only by changes in temperature.

Compressibility: k=8.50*10^-10 [Pa^-1]

β(Liquid)=4.80*10^-4

β(Metal)=3.90*10^-5

**2. Homework Equations**

Bulk Modulus: B= Δp/(ΔV/V)

Linear Volume expansion: ΔV=β*ΔT*V

Compressibility: k=1/B

**3. The Attempt at a Solution**

So using the bulk modulus definition I found an expression that relates changes in pressure to changes in temperature. With this I found for the liquid in the system the change in temperature is equal to 8.97 C. Meaning the temperature of the liquid after compression is 38.97 C.

I understand that a change in pressure of the system leads to a change in temperature of the system; which in turn leads to a change in volume of the systems components. The way I'm seeing the problem now is now that I know the temperature of the liquid after compression I can assume the temperature of the container is the same.

The pressure is 51 atm after compression, therefore to get it back to 1 atm we must change the temperature of the system to account for a pressure change of -50 atm. But if I do this I'm literally just reversing what I did in the last step. I don't know what I'm missing. I know I'm not using the expansion coefficient of the metal in any way despite that it was given so it probably involves that.