Question

A piston-cylinder device initially contains \(0.4 \mathrm{kg}\) of nitrogen gas at \(160 \mathrm{kPa}\) and \(140^{\circ} \mathrm{C}\). The nitrogen is now expanded isothermally to a pressure of 100 kPa. Determine the boundary work done during this process.

Short answer

Based on the given information, the boundary work done during the isothermal expansion of nitrogen gas inside the piston-cylinder device is approximately 37,460 J.

Step-by-step solution

Find the initial volume

We are given the initial pressure, temperature, and mass of the nitrogen gas. We can find the initial volume using the ideal gas law: \(PV = nRT\) First, we need to convert mass to moles and the temperature to Kelvin. The molar mass of nitrogen is 28.02 g/mol: \(n = \dfrac{mass}{molar \thinspace mass} = \dfrac{0.4 kg}{28.02 \frac{g}{mol}} \cdot \dfrac{1000}{1} = 14.275 \thinspace mol\) Next, we convert the temperature to kelvin: \(T = 140 + 273.15 = 413.15 K\) Now, we can find the initial volume using the ideal gas law: \(V_1 = \dfrac{nRT}{P_1} = \dfrac{(14.275 \thinspace mol)(8.314 \frac{J}{mol \cdot K})(413.15 K)}{160 kPa} = 0.478 \thinspace m^3\)

Find the final volume

Since the process is isothermal, we can use the same temperature and the final pressure to find the final volume of the nitrogen gas: \(V_2 = \dfrac{nRT}{P_2} = \dfrac{(14.275 \thinspace mol)(8.314 \frac{J}{mol \cdot K})(413.15 K)}{100 kPa} = 0.765 \thinspace m^3\)

Calculate the boundary work

When the gas expands isothermally, the boundary work done can be calculated using the following formula: \(W_{boundary} = nRT \ln{\dfrac{V_2}{V_1}}\) Plugging in the values from steps 1 and 2, we get: \(W_{boundary} = (14.275 \thinspace mol)(8.314 \frac{J}{mol \cdot K})(413.15 K) \ln{\dfrac{0.765 \thinspace m^3}{0.478 \thinspace m^3}}\) \(W_{boundary} \approx 37460 \thinspace J\) So, the boundary work done during this isothermal expansion is approximately \(37,460 J\).