To determine the work necessary to compress air in an isolated cylinder, we need to consider the ideal gas law and the process of compression.
The ideal gas law relates the pressure, volume, and temperature of an ideal gas and can be expressed as:
PV = nRT,
where: P is the pressure of the gas, V is the volume of the gas, n is the number of moles of gas, R is the ideal gas constant, T is the temperature of the gas in Kelvin.
First, we need to convert the initial temperature from Celsius to Kelvin:
T_initial = 20 + 273.15 = 293.15 K.
Next, we can determine the number of moles of air using the ideal gas law at the initial conditions:
n_initial = (P_initial * V_initial) / (R * T_initial),
where: P_initial = 200 kPa = 200,000 Pa, V_initial = 2 m^3, R = 8.314 J/(mol·K).
Substituting the values:
n_initial = (200,000 * 2) / (8.314 * 293.15) ≈ 161.67 mol.
Now, let's determine the final pressure and temperature after compression. Since the process is not specified, we assume it to be an isothermal process, meaning the temperature remains constant during compression.
The ideal gas law can be rearranged as:
P_final * V_final = n_initial * R * T_final.
Since it's an isothermal process, T_final = T_initial = 293.15 K.
Now, we can solve for P_final using the given V_final = 0.2 m^3:
P_final = (n_initial * R * T_final) / V_final,
P_final = (161.67 * 8.314 * 293.15) / 0.2 ≈ 6,723,945 Pa.
To calculate the work done during compression, we need to consider that work is given by the area under the pressure-volume (P-V) curve. In an isothermal process, the work can be calculated as:
Work = -n_initial * R * T_initial * ln(V_final / V_initial),
where ln represents the natural logarithm.
Substituting the values:
Work = -161.67 * 8.314 * 293.15 * ln(0.2 / 2) ≈ -129,930 J.
Note that the work is negative, indicating that work is done on the system during compression.
Therefore, the work necessary to compress the air in the isolated cylinder from a volume of 2 m^3 to 0.2 m^3 is approximately -129,930 J (or -129.93 kJ).