0.15m3 of air, initially at a pressure of 100kN/m2 and temperature of 20°C iscompressed to a pressure of 1.5 MN/m2 according to the law p.V1.2=C.Find:i) The work done during compression. Is the work done on or by the gas?
Question
0.15m3 of air, initially at a pressure of 100kN/m2 and temperature of 20°C iscompressed to a pressure of 1.5 MN/m2 according to the law p.V1.2=C.Find:i) The work done during compression. Is the work done on or by the gas?
Solution 1
To solve this problem, we need to use the formula for work done during an isentropic process, which is given by:
W = (P2V2 - P1V1) / (n-1)
where: P1 = Initial pressure = 100 kN/m^2 = 100,000 N/m^2 V1 = Initial volume = 0.15 m^3 P2 = Final pressure = 1.5 MN/m^2 = 1,500,000 N/m^2 n = polytropic index = 1.2 (from the law p.V^n = C)
We need to find V2 (final volume) first. We can find it using the polytropic process equation:
P1V1^n = P2V2^n
=> V2 = (P1/P2)^(1/n) * V1 => V2 = (100,000/1,500,000)^(1/1.2) * 0.15 => V2 = 0.056 m^3
Now, we can find the work done:
W = (P2V2 - P1V1) / (n-1) => W = (1,500,0000.056 - 100,0000.15) / (1.2-1) => W = 2.1 MJ
The work is positive, which means it is done on the gas. This is because we are compressing the gas, which requires work to be done on it.
Solution 2
To solve this problem, we need to use the formula for work done on a gas during compression or expansion, which is given by:
W = ∫PdV
However, since the pressure is not constant, we need to express P in terms of V. Given the law p.V^1.2 = C, we can express P as:
P = C/V^1.2
Substituting this into the work formula gives:
W = ∫(C/V^1.2) dV
We need to integrate this from the initial volume (V1 = 0.15 m^3) to the final volume (V2), which we don't know yet. But we can find V2 using the given final pressure (P2 = 1.5 MN/m^2) and the law p.V^1.2 = C. Since C is constant, we have:
P1.V1^1.2 = P2.V2^1.2
Substituting the given values (P1 = 100 kN/m^2, V1 = 0.15 m^3, P2 = 1.5 MN/m^2) gives:
(100 kN/m^2 * (0.15 m^3)^1.2) = 1.5 MN/m^2 * V2^1.2
Solving this equation gives the final volume V2. Once we have V2, we can compute the work W by evaluating the integral ∫(C/V^1.2) dV from V1 to V2.
Finally, the work is done on the gas because the gas is being compressed.
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