Based on the available information, the work for the process is approximately -30,030 Joules. The negative sign indicates that work is done on the system.
To determine the work for the process, we can use the formula for work done in a piston-cylinder assembly:
W = ∫(PdV)
Given that the relationship between pressure and volume is pV² = constant, we can rewrite it as:
P = k / V²
where k is the constant.
To find the value of k, we can use the initial conditions:
P1 = 1 bar = 100,000 Pa
V1 = 0.1 m^3
Substituting these values into the equation, we get:
100,000 = k / (0.1)²
k = 100,000 * (0.1)²
k = 100,000 * 0.01
k = 1,000
Now, we can express the pressure as a function of volume:
P = 1,000 / V²
To calculate the work, we integrate the expression for work:
W = ∫(PdV) = ∫((1,000 / V²)dV)
Integrating this expression, we get:
W = -1,000 / V
To find the limits of integration, determine the final volume (V2) corresponding to the final pressure:
P2 = 9 bar = 900,000 Pa
Substituting this value into the equation, we get:
900,000 = 1,000 / V2²
V2² = 1,000 / 900,000
V2² = 0.001111...
V2 ≈ 0.0333 m³
Now, we can calculate the work:
W = -1,000 / V
W = -1,000 / 0.0333
W ≈ -30,030 J
Therefore, the work for the process is approximately -30,030 Joules. The negative sign indicates that work is done on the system.