A mass of gas at an initial pressure of n28 bar and with an internal energyof 1500kJ is contained in a well insulated cylinder of volume 0.06m3. The gasis allowed to expand until its internal energy is 1400kJ, with law of expansionbeing pv2=c. Calculate work done and final volume & pressure of the gas.
Question
A mass of gas at an initial pressure of n28 bar and with an internal energyof 1500kJ is contained in a well insulated cylinder of volume 0.06m3. The gasis allowed to expand until its internal energy is 1400kJ, with law of expansionbeing pv2=c. Calculate work done and final volume & pressure of the gas.
Solution
The problem involves the expansion of a gas in a well-insulated cylinder, which means it's an adiabatic process. The law of expansion is given as pv^2 = c, which is a polytropic process where n = 2.
Step 1: Calculate the constant, c, using the initial conditions.
The initial pressure, p1 = 28 bar = 28 x 10^5 Pa The initial volume, v1 = 0.06 m^3 So, c = p1 * (v1)^2 = 28 x 10^5 * (0.06)^2 = 100800 Pa.m^6
Step 2: Calculate the work done during the process.
The work done (W) in an adiabatic process is equal to the change in the internal energy of the system. So, W = ΔU = U1 - U2 = 1500 kJ - 1400 kJ = 100 kJ = 100 x 10^3 J
Step 3: Calculate the final volume, v2.
We know that the internal energy of an ideal gas is directly proportional to its temperature and hence, the volume (since the process is adiabatic). Therefore, the ratio of the initial and final volumes is equal to the ratio of the initial and final internal energies. So, v2 = v1 * (U2/U1) = 0.06 m^3 * (1400 kJ / 1500 kJ) = 0.056 m^3
Step 4: Calculate the final pressure, p2.
We can use the polytropic process equation, pv^2 = c, to find the final pressure. So, p2 = c / (v2)^2 = 100800 Pa.m^6 / (0.056 m^3)^2 = 32.14 x 10^5 Pa = 32.14 bar
So, the work done is 100 kJ, the final volume is 0.056 m^3, and the final pressure is 32.14 bar.
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