Question

Helium contained in a closed, rigid tank, initially at 50°C, 2 bar, and a volume of 2 m³, is heated to a final pressure of 8 bar. Assume the ideal gas model for the helium. Kinetic and potential energy effects can be ignored.

​
​Determine the mass of the helium, in kg, and the heat transfer, in kJ.

For an ideal gas in a closed system undergoing a constant volume process:

Option A: the difference of (final – initial) temperature equals the difference of (initial – final) pressure.

Option B: the ratio of final/initial temperature equals the ratio of final/initial pressure.

Option C: the difference of (final – initial) temperature equals the difference of (final – initial) pressure.

Option D: the ratio of final/initial temperature equals the ratio of initial/final pressure.

Answer 1

The mass and heat transfer of helium in a closed, rigid container undergoing a constant volume heating process can be determined using the ideal gas law and the concept of internal energy change. The mass is calculated from initial conditions using the equation PV= mRT, and the heat transfer is determined based on the change in internal energy, ΔU, which equals the heat transfer (Q) in a constant volume process. Option D correctly describes the relationship between temperature and pressure changes in this scenario.

Step-by-step explanation:

The question relates to a closed system containing helium, which is considered as an ideal gas and is undergoing a constant volume heating process. To find the mass of the helium, we use the ideal gas equation PV = mRT, where P is the pressure, V is the volume, m is the mass, R is the specific gas constant for helium, and T is the temperature in Kelvin.

Firstly, we convert the initial temperature from Celsius to Kelvin (50°C + 273.15 = 323.15 K). Then we calculate the mass (m) of the helium using the initial conditions:

m = PV / RT = (2 bar * 2 m3) / (R * 323.15 K)

Note that 1 bar equals 100,000 Pa, and helium's specific gas constant (R) is 2077 J/(kg·K). This gives us the mass of helium.

To determine the heat transfer, we start by finding the final temperature using the combined gas law for the constant volume process:

P1/T1 = P2/T2 -> T2 = P2 * T1 / P1

Once T2 is found, the change in internal energy (ΔU) of the gas can be calculated using ΔU = mCvΔT, where Cv is the specific heat capacity at constant volume for helium (3/2 R per mole). The heat transfer (Q) is equal to ΔU since we are assuming no work done in a constant volume process (W=0).

The correct option regarding the changes in temperature and pressure of an ideal gas in a constant volume process is Option D: the ratio of final/initial temperature equals the ratio of initial/final pressure.