Question

One kilogram of carbon dioxide is compressed from \(0.5 \mathrm{MPa}\) and \(200^{\circ} \mathrm{C}\) to \(3 \mathrm{MPa}\) in a piston-cylinder device arranged to execute a polytropic process with \(n=1.3 .\) Use the compressibility factor to determine the final temperature.

Short answer

Answer: The final temperature of the carbon dioxide is approximately 709.68 °C.

Step-by-step solution

Determine the initial molar volume

To determine the initial molar volume (V1), we can use the initial pressure (P1) and temperature (T1) values given along with the ideal gas law. The molecular weight of Carbon Dioxide (CO2), M = 44 g/mol. Given, mass of CO2, m = 1 kg. Now, convert the mass to moles: n = m / M = (1 * 1000) / 44 = 22.73 mol. T1 = 200 + 273.15 = 473.15 K (convert from Celsius to Kelvin) P1 = 0.5 MPa = 0.5 * 10^6 Pa Now, use the ideal gas law to find V1: PV = nRT V1 = (nRT) / P1 Using the gas constant R = 8.314 J / (mol*K) V1 = (22.73 * 8.314 * 473.15) / (0.5 * 10^6) = 0.1845 m^3/mol

Find the initial compressibility factor and specific volume

We have to find the initial compressibility factor (Z1) for CO2 at the given initial pressure and temperature. The compressibility factor can often be found in a chart or provided in an exercise. For the given initial conditions, let's assume Z1 = 0.85. Initial specific volume, v1 = V1 / n = 0.1845 / 22.73 = 0.00812 m^3/kg.

Use the polytropic process equation to find the final specific volume

The polytropic process equation is given by: (P1 * v1^n = P2 * v2^n) Rearrange this equation to solve for v2: v2 = (P1 * v1^n) / P2 Given, P2 = 3 MPa = 3 * 10^6 Pa v2 = (0.5 * 10^6 * (0.00812)^1.3) / (3 * 10^6) = 0.00273 m^3/kg

Calculate the final compressibility factor and get the final pressure

Calculate the final molar volume (V2) using the final specific volume (v2) and moles (n): V2 = n * v2 = 22.73 * 0.00273 = 0.0620 m^3/mol Find the final compressibility factor (Z2) for CO2 at the given final pressure and unknown final temperature. Let's assume Z2 = 0.87 as an approximation. Now, use the ideal gas law to find the final pressure (P2): P2 = (n * Z2 * R * T2) / V2 Rearrange to get T2: T2 = (P2 * V2) / (n * Z2 * R)

Use the ideal gas law to find the final temperature

Plug the values into the equation to find the final temperature, T2: T2 = (3 * 10^6 * 0.0620) / (22.73 * 0.87 * 8.314) = 982.83 K Convert the temperature back to Celsius: T2 = 982.83 - 273.15 = 709.68 °C The final temperature of the carbon dioxide is approximately 709.68 °C.