Question

Air is contained in a cylinder device fitted with a piston-cylinder. The piston initially rests on a set of stops, and a pressure of \(200 \mathrm{kPa}\) is required to move the piston. Initially, the air is at \(100 \mathrm{kPa}\) and \(23^{\circ} \mathrm{C}\) and occupies a volume of \(0.25 \mathrm{m}^{3} .\) Determine the amount of heat transferred to the air, in \(\mathrm{kJ},\) while increasing the temperature to \(700 \mathrm{K}\). Assume air has constant specific heats evaluated at \(300 \mathrm{K}\).

Short answer

Answer: The heat transfer required to increase the temperature to 700 K is approximately 1,180.2 kJ.

Step-by-step solution

Convert initial temperature to Kelvin

First, to work with specific heats, we want to convert the initial temperature given in Celsius to Kelvin. We do this by adding 273 to the temperature in Celsius: \(T_1 = 23^{\circ}C + 273 = 296K\)

Calculate the mass of the air

We will use the Ideal Gas Law to find the mass of the air inside the cylinder: \(PV = mRT\) Where \(P\) is the pressure, \(V\) is the volume, \(m\) is the mass, \(R\) is the specific gas constant, and \(T\) is the temperature. Rearranging for mass, we get: \(m = \frac{PV}{RT}\) For air, the specific gas constant is \(R = 287 \frac{J}{kg \cdot K}\). The volume is given as \(0.25m^3\) and the pressure is 100 kPa, so we can plug these values into the equation above: \(m = \frac{100 \times 10^3 \times 0.25}{287 \times 296} = 2.9065 kg\)

Calculate final pressure and volume

We are assuming an isobaric process, so the final pressure will be the same as the initial pressure: \(P_2 = P_1 = 100 \mathrm{kPa}\) To find the final volume, we can use the Ideal Gas Law again: \(V_2 = \frac{mRT_2}{P_2} = \frac{(2.9065)(287)(700)}{100 \times 10^3} = 0.5102 \mathrm{m}^3\)

Calculate heat transfer

For an isobaric process, the heat transfer can be calculated using the formula below: \(Q = mc(T_2 - T_1)\) The specific heat at constant pressure for air at 300 K is \(c_p = 1005 \frac{J}{kg \cdot K}\). Now, we can plug the values into the formula: \(Q = (2.9065)(1005)(700 - 296) = 2.9065 \times 1005 \times 404 = 1180.2 kJ\) So, the amount of heat transferred to the air while increasing the temperature to \(700K\) is approximately \(1,180.2 kJ\).

Key concepts

Ideal Gas Law

The Ideal Gas Law is fundamental to understanding how gases behave under various conditions of pressure, volume, and temperature. It is often expressed by the equation:
\(PV = nRT\),
where \(P\) stands for pressure, \(V\) is volume, \(n\) corresponds to the number of moles of the gas, \(R\) is the universal gas constant, and \(T\) denotes temperature in kelvins. In the context of the exercise given to us, the Ideal Gas Law is rearranged to find the mass \(m\) of the air, by using the specific gas constant \(R\) for air and incorporating density. This step is crucial because it sets the stage to analyze how much heat needs to be transferred to the air in the piston to raise its temperature.

In practice, remember to always work with Kelvin temperatures when using the Ideal Gas Law, as it is a temperature scale that begins at absolute zero. This law is particularly useful because it adapts to changes in a gas’s condition, which allows us to predict one variable, if the other three are known.

Isobaric Process

In thermodynamics, an isobaric process is one that occurs at a constant pressure. The term 'isobaric' derives from the Greek words 'iso' meaning 'equal' and 'baros' meaning 'pressure'. During an isobaric process, even as the volume and temperature of a gas may change, the pressure remains constant.

The significance of the isobaric process in our given exercise is that it simplifies the calculation of heat transfer. This is because the work done by or on the system can be easily computed when pressure is constant. By keeping the pressure constant in the exercise, we are able to directly use the specific heat at constant pressure (\(c_p\)) in the formula to calculate the heat added or removed, without having to correct for changes in pressure. This gives a direct relationship between heat transfer and the temperature change, allowing us to find the total heat transferred to the air as it is heated.

Specific Heats

Specific heats are properties that dictate how much heat energy is required to raise the temperature of a substance by a certain amount. There are two kinds of specific heats: \(c_p\), the specific heat at constant pressure, and \(c_v\), the specific heat at constant volume. For an isobaric process, we use \(c_p\) because the process occurs at a fixed pressure.

Specific heats are measured in units of energy per unit mass per degree (usually in Joules per kilogram per Kelvin, \(J/(kg⋅K)\)). In the scenario of the exercise, knowing the specific heat at constant pressure of the air (assuming it's at room temperature) is crucial to determining how much heat energy is transferred to the air. This allows us to calculate the heat transfer by simply multiplying the mass of the air, its specific heat capacity, and the change in temperature, as shown in the formula:
\(Q = mc_p(T_2 - T_1)\).
This formula relies on the fact that the specific heats are constant values for the given temperature range, which in real-life applications, might vary with temperature. Therefore, assuming a constant specific heat can lead to slight inaccuracies, but it serves as a good approximation for many engineering purposes.