a perfectly insulated rigid tank with a volume of 0.2metre cube contains a perfect gas which has a molar mass of 18kg per mole and ratio of specific heats of 1.45. initially the pressure and temperature in the tank are 9 bar and 320k respectively. A fan inside the tank is span at 360 revolutions per min for 20 seconds. The torque required to turn a fan is 30Nm. calculate the R, cp and cv values of the gad and the masd of the gas in the tank

To tackle this problem, we need to break it down into several parts: calculating the gas constant (R), the specific heat capacities at constant pressure (Cp) and constant volume (Cv), and finally determining the mass of the gas in the tank. Let's go through each step methodically.

1. Calculating the Gas Constant (R)

The gas constant (R) can be calculated using the formula:

R = R_u / M

Where:

• R_u is the universal gas constant, approximately 8.314 J/(mol·K).
• M is the molar mass of the gas in kg/mol.

Given that the molar mass (M) is 18 kg/kmol (or 0.018 kg/mol), we can substitute this value into the equation:

R = 8.314 J/(mol·K) / 0.018 kg/mol

Calculating this gives:

R ≈ 461.67 J/(kg·K)

2. Finding Specific Heat Capacities (Cp and Cv)

To find the specific heat capacities, we can use the relationship between them and the ratio of specific heats (γ):

γ = Cp / Cv

Given that γ = 1.45, we can express Cp in terms of Cv:

Cp = γ * Cv

We also know the relationship between Cp, Cv, and R:

Cp - Cv = R

Substituting Cp from the first equation into the second gives:

γ * Cv - Cv = R

This simplifies to:

(γ - 1) * Cv = R

Now, substituting the known values:

(1.45 - 1) * Cv = 461.67

0.45 * Cv = 461.67

Solving for Cv:

Cv ≈ 1025.93 J/(kg·K)

Now, substituting Cv back to find Cp:

Cp = 1.45 * 1025.93 ≈ 1486.60 J/(kg·K)

3. Determining the Mass of the Gas in the Tank

The mass of the gas can be calculated using the ideal gas law:

P * V = m * R * T

Where:

• P is the pressure in pascals (9 bar = 900,000 Pa).
• V is the volume (0.2 m³).
• m is the mass of the gas.
• R is the specific gas constant we calculated earlier.
• T is the temperature in Kelvin (320 K).

Rearranging the equation to solve for mass (m) gives:

m = (P * V) / (R * T)

Substituting the values:

m = (900,000 Pa * 0.2 m³) / (461.67 J/(kg·K) * 320 K)

Calculating this yields:

m ≈ 2.52 kg

Summary of Results

To summarize our findings:

• Gas Constant (R): 461.67 J/(kg·K)
• Specific Heat Capacity at Constant Pressure (Cp): 1486.60 J/(kg·K)
• Specific Heat Capacity at Constant Volume (Cv): 1025.93 J/(kg·K)
• Mass of the Gas in the Tank: 2.52 kg

This comprehensive approach allows us to understand the thermodynamic properties of the gas contained in the tank, as well as the implications of the fan's operation within that system.