Question

Helium gas is compressed from \(27^{\circ} \mathrm{C}\) and \(3.50 \mathrm{m}^{3} / \mathrm{kg}\) to \(0.775 \mathrm{m}^{3} / \mathrm{kg}\) in a reversible and adiabatic manner. The temperature of helium after compression is \((a) 74^{\circ} \mathrm{C}\) \((b) 122^{\circ} \mathrm{C}\) \((c) 547^{\circ} \mathrm{C}\) \((d) 709^{\circ} \mathrm{C}\) \((e) 1082^{\circ} \mathrm{C}\)

Short answer

a) 300 °C b) 450 °C c) 547 °C d) 600 °C Given: Initial temperature: T1 = 300 K Initial specific volume: v1 = 3.50 m³/kg Final specific volume: v2 = 0.775 m³/kg Specific heat ratio for helium: γ = 5/3 Solution: 1. Calculate the final temperature (T2) using the adiabatic relation for an ideal gas: T2 = T1(v1/v2)^(γ - 1) 2. Substitute the given values and solve for T2: T2 ≈ 842 K 3. Convert the final temperature to Celsius: T2°C ≈ 569 °C Answer: The closest option to the calculated final temperature is (c) 547 °C.

Step-by-step solution

Write down the given information

The initial temperature is \(27^{\circ} \mathrm{C}\), which in Kelvin is \(T_1 = 300\mathrm{K}\). The initial specific volume is \(v_1 = 3.50 \mathrm{m}^{3} / \mathrm{kg}\). The final specific volume is \(v_2 = 0.775 \mathrm{m}^{3} / \mathrm{kg}\). We also know that the process is adiabatic and reversible.

Find the specific heat ratio for helium

For helium, the specific heat ratio is \(\gamma = \frac{C_P}{C_V} = \frac{5}{3}\).

Apply the adiabatic relation for an ideal gas

The adiabatic relation for an ideal gas is given by: \((\frac{T_1}{T_2})^{\gamma - 1} = \frac{v_2}{v_1}\) We need to find \(T_2\), so we should rearrange the equation to make \(T_2\) the subject: \(T_2 = T_1(\frac{v_1}{v_2})^{\gamma - 1}\)

Substitute the known values and solve for the final temperature

Now, we can substitute the known values into the equation: \(T_2 = 300\mathrm{K}(\frac{3.50 \mathrm{m}^{3} / \mathrm{kg}}{0.775 \mathrm{m}^{3} / \mathrm{kg}})^{\frac{5}{3} - 1}\) \(T_2 \approx 842\mathrm{K}\)

Convert the temperature to Celsius and compare it to the given options

The final temperature in Celsius is \(T_2^{\circ} \mathrm{C} = 842\mathrm{K} - 273.15 \approx 569^{\circ} \mathrm{C}\). Comparing this to the given options, we can see that the closest option is \((c) 547^{\circ} \mathrm{C}\), so the answer is \((c) 547^{\circ} \mathrm{C}\).