Question

A gas in a piston–cylinder assembly undergoes a compression process for which the relation between pressure and volume is given by pVn = constant. The initial volume is 0.1 m^3 , the final volume is 0.04 m^3 , and the final pressure is 2 bar. Determine the initial pressure, in bar, and the work for the process, in kJ, if (a) n = 0, (b) n = 1, (c) n = 1.3

Answer 1

(a) The final pressure is 2 bar.

(b) n = 1, pressure is 1.6 kJ.

(c)The initial pressure is ≈ 1.84 kJ.

Step-by-step explanation:

To determine the initial pressure and work for the compression process, we need to use the relation between pressure and volume given by pV^n = constant.

a) When n = 0, the relation becomes pV^0 = constant, which means the pressure is constant throughout the process. Since the final pressure is given as 2 bar, the initial pressure is also 2 bar.

b) When n = 1, the relation becomes pV = constant. Using the initial and final volumes, we can calculate the initial pressure: p * 0.1 = 2 * 0.04. Solving for p, we find p = 0.32 bar. The work done can be calculated using the formula W = -∆U, where ∆U is the change in internal energy. Since the process is defined as compression, ∆U will be negative. The work can then be calculated using the ideal gas law equation: W = nRT * ln(V_f / V_i). Plugging in the values, we get W = 1.6 kJ.

c) When n = 1.3, the relation becomes pV^1.3 = constant. Using the initial and final volumes, we can calculate the initial pressure: p * 0.1^1.3 = 2 * 0.04^1.3. Solving for p, we find p ≈ 1.59 bar. The work can be calculated using the same formula as in part b: W = nRT * ln(V_f / V_i). Plugging in the values, we get W ≈ 1.84 kJ.