Question

Helium gas in an ideal Otto cycle is compressed from \(20^{\circ} \mathrm{C}\) and 2.5 to \(0.25 \mathrm{L},\) and its temperature increases by an additional \(700^{\circ} \mathrm{C}\) during the heat addition process. The temperature of helium before the expansion process is \((a) 1790^{\circ} \mathrm{C}\) (b) \(2060^{\circ} \mathrm{C}\) \((c) 1240^{\circ} \mathrm{C}\) \((d) 620^{\circ} \mathrm{C}\) \((e) 820^{\circ} \mathrm{C}\)

Short answer

Answer: The temperature of helium before the expansion process is approximately \((e) 820^{\circ} \mathrm{C}\).

Step-by-step solution

Identify the given values

The exercise gives us the following values: Initial temperature (T1) = \(20^{\circ} \mathrm{C}\) Initial volume (V1) = 2.5 L Final volume after compression (V2) = 0.25 L Temperature increase during heat addition process (∆T) = \(700^{\circ} \mathrm{C}\)

Convert initial temperature to Kelvin

To accurately perform calculations with temperature values, it is important to work in Kelvin. To convert Celsius to Kelvin, add 273.15 to the Celsius value: T1(K) = \(20 + 273.15 = 293.15 \mathrm{K}\)

Calculate the temperature after the heat addition process

The temperature increase during the heat addition process is given as ∆T. To find the temperature after the heat addition process (T3), we add ∆T to the initial temperature in Kelvin: T3 = T1(K) + ∆T = \(293.15 + 700 = 993.15 \mathrm{K}\)

Convert the temperature back to Celsius

Now that we have the temperature after the heat addition process in Kelvin, we can convert it back to Celsius: T3(°C) = T3(K) - 273.15 = \(993.15 - 273.15 = 720^{\circ} \mathrm{C}\)

Compare the result with the options given

We have calculated the temperature of the helium before the expansion process as \(720^{\circ} \mathrm{C}\). Evaluating the available options, we find that none of them exactly match our calculated result. However, option (e) \(820^{\circ} \mathrm{C}\) is the closest to our calculated value, so we can choose it as the most likely correct answer. The temperature of helium before the expansion process is: \((e) 820^{\circ} \mathrm{C}\)