Question

Air is contained in a piston-cylinder device at \(600 \mathrm{kPa}\) and \(927^{\circ} \mathrm{C},\) and occupies a volume of \(0.8 \mathrm{m}^{3} .\) The air undergoes and isothermal (constant temperature) process until the pressure in reduced to 300 kPa. The piston is now fixed in place and not allowed to move while a heat transfer process takes place until the air reaches \(27^{\circ} \mathrm{C}\) (a) Sketch the system showing the energies crossing the boundary and the \(P\) - \(V\) diagram for the combined processes. (b) For the combined processes determine the net amount of heat transfer, in \(\mathrm{kJ},\) and its direction. Assume air has constant specific heats evaluated at \(300 \mathrm{K}\).

Short answer

a) On the P-V diagram, the initial isothermal process will be represented by a curve that expands horizontally as the volume increases while the pressure decreases. The energy crossing the boundary during this process includes positive work done by the gas (expansion) and positive heat transfer (from the surroundings to the system). The second process, constant volume, will be shown as a vertical line where pressure decreases, and no work is done since the volume is constant. However, negative heat transfer occurs (from the system to the surroundings) during this process. b) Without the value of the specific heat at constant volume (C_v), the exact net amount of heat transfer cannot be calculated. However, we can infer that the first process has positive heat transfer (from the surroundings to the system), while the second process has negative heat transfer (from the system to the surroundings). The overall direction of the net heat transfer will depend on which process has a larger magnitude of heat transfer.

Step-by-step solution

(a) Isothermal process initial parameters

Convert the initial temperature to Kelvin and calculate the initial amount of substance (moles) using the Ideal Gas Law. Initial temperature, \(T_1 = 927 + 273.15 = 1200.15 \mathrm{K}\) Initial volume, \(V_1 = 0.8 \mathrm{m^3}\) Initial pressure, \(P_1 = 600 \mathrm{kPa} = 600000 \mathrm{Pa}\) Air is assumed to behave as an ideal gas with a molar gas constant, \(R = 8.314 \mathrm{J}\cdot\mathrm{mol}^{-1}\cdot\mathrm{K}^{-1}\). Using the Ideal Gas Law, we find the amount of substance n: \(n = \frac{PV}{RT} = \frac{600000 \cdot 0.8}{8.314 \cdot 1200.15} \approx 24.24 \mathrm{mol}\)

(a) Isothermal process final parameters, work, and heat transfer

For the final state in the isothermal process, first, calculate the final volume using the initial temperature and pressure. Final pressure, \(P_2 = 300 \mathrm{kPa} = 300000 \mathrm{Pa}\) In an isothermal process, \(T_1 = T_2\). So, Final volume, \(V_2 = \frac{nRT_2}{P_2} = \frac{24.24 \cdot 8.314 \cdot 1200.15}{300000} \approx 1.6 \mathrm{m^3} \) Calculate work done by the gas during the isothermal process using the equation for isothermal expansion: \(W = nRT\ln{\frac{V_2}{V_1}}\) \(W = (24.24)(8.314)(1200.15)\ln\frac{1.6}{0.8} \approx 24316.37 \mathrm{J} = 24.316 \mathrm{kJ}\) The sign of work is positive, which means that the gas has done work on the surroundings during the expansion. Calculate the heat transfer during the isothermal process: Since this is an isothermal process, the internal energy change is 0, and the heat transfer is equivalent to the work done by the gas: \(Q = W = 24.316 \mathrm{kJ}\)

(a) Constant volume process parameters, and heat transfer

For the constant volume process, first, calculate the final temperature in Kelvin. Final temperature, \(T_3 = 27 + 273.15 = 300.15 \mathrm{K}\) Calculate the heat transfer during the constant volume process: Since the piston is fixed, no work is done by the gas, and the heat transfer is equivalent to the change in internal energy of the gas. We can use the equation: \(Q = nC_v(T_3 - T_2)\), assuming air has constant specific heat at constant volume, \(C_v\) \(Q = (24.24)(C_v)(300.15 - 1200.15) \mathrm{kJ} \)

(b) Calculation of the net amount of heat transfer in the combined processes

To calculate the net amount of heat transfer, we need to add the heat transfer in the isothermal process (\(Q_{iso}\)) and the heat transfer in the constant volume process (\(Q_{cv}\)): \(Q_{net} = Q_{iso} + Q_{cv}\) \(Q_{net} = 24.316 + (24.24)(C_v)(300.15 - 1200.15) \mathrm{kJ}\) Without the value of specific heat at constant volume (\(C_v\)), it is impossible to find the exact numerical value of the net heat transfer. However, knowing that \(C_v > 0\), we can conclude that the heat transfer in the constant volume process is negative and that the value of the net heat transfer will depend on this process. Finally, the direction of the net amount of heat transfer will be determined by the sign of \(Q_{net}\). Currently, we cannot find the exact numerical value as \(C_v\) is unknown. However, we do know that the first process has positive heat transfer (from the surroundings to the system) and the second process has negative heat transfer (from the system to the surroundings). The overall direction will depend on which process has a larger magnitude of heat transfer.

Key concepts

Isothermal Process

An isothermal process is a transformation in a system where the temperature remains constant. This term comes from the Greek words 'iso', meaning equal, and 'thermos', meaning heat. During an isothermal process in thermodynamics, the system either absorbs or releases heat in such a manner that the temperature within the system does not change.

For an ideal gas, when an isothermal process occurs – like in the scenario where air is compressed or expanded within a piston-cylinder device – the heat transfer into or out of the system results in work being done but leaves the internal energy of the gas unchanged. This concept is showcased when we see the air undergo an isothermal expansion: the gas does work on the surroundings, with the equivalence of heat being absorbed to maintain a steady temperature.

To understand the process better, envision the amount of heat absorbed by the gas as equal to the amount of work done during expansion. This is why during an isothermal process that conforms to the ideal gas law, the heat transfer (Q) can be calculated using the work done by the gas (W), which stands as Q = W, assuming no change in internal energy.

Ideal Gas Law

The ideal gas law is a fundamental equation in the study of thermodynamics, universally expressed as PV = nRT, where P is the pressure, V is the volume, n is the number of moles, R is the universal gas constant, and T is the absolute temperature in Kelvin.

The ideal gas law represents a hypothetical gas that perfectly follows this linear relationship among its pressure, volume, and temperature, with no deviations. In real-world contexts, most gases at high temperatures and low pressures mimic the behavior predicted by the ideal gas law closely enough for it to be a useful approximation.

By calculating the number of moles (n) using this law, as seen in the provided solution, we understand the molar quantity of the gas which we can then use to deduce other thermodynamic properties of the system such as the work done during processes or the internal energy changes.

PV Diagram

A pressure-volume (PV) diagram is a graphical tool used in thermodynamics to visualize the changes that a gaseous system undergoes during a thermodynamic process. The axes represent the pressure ((P) and volume ((V), and the state of the gas at any point in the process can be plotted as a coordinate in this space.

The PV diagram enables us to spatially envision the relationship between pressure and volume during processes, such as isothermal (constant temperature) or isochoric (constant volume) processes. When sketching the diagram, an isothermal process at constant temperature appears as a hyperbolic curve, while a constant volume process appears as a vertical line, suggesting no change in volume.

Using the PV diagram in thermodynamics allows students and engineers alike to determine work done by or on the system by finding the area under the process curve on the diagram.

Specific Heats

Specific heats are properties of substances that dictate how much heat energy is required to raise the temperature of a unit mass of the substance by one degree Celsius. They are critical variables in the study of thermodynamics and heat transfer. There are two kinds of specific heats: at constant volume (C_v), and at constant pressure (C_p).

These properties are crucial for determining heat transfer associated with temperature changes when the volume or pressure remains constant. For ideal gases, these specific heats are related, but they remain distinct properties with different values. Using the specific heat in calculations, as presented in the exercise, lets us quantify the heat transferred when the air within the cylinder is no longer allowed to expand and is cooled at a constant volume. A constant C_v value implies that the specific heat does not change with temperature, which is a close approximation for ideal gases over a limited temperature range.