To tackle this problem, we need to break it down into several parts: calculating the specific 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 Specific Gas Constant (R)
The specific gas constant (R) can be calculated using the formula:
R = R₀ / M
Where:
- R₀ 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) of the gas is 18 kg/mol, we can substitute this value into the equation:
R = 8.314 J/(mol·K) / 18 kg/mol
Calculating this gives:
R ≈ 0.461 J/(kg·K)
2. Determining Cp and Cv
For a perfect gas, the relationship between Cp, Cv, and R is given by:
Cp - Cv = R
Additionally, the ratio of specific heats (γ) is defined as:
γ = Cp / Cv
Given that γ = 1.45, we can express Cp and Cv in terms of Cv:
Cp = γ * Cv
Substituting this into the first equation gives:
γ * Cv - Cv = R
This simplifies to:
(γ - 1) * Cv = R
Now, substituting the known values:
(1.45 - 1) * Cv = 0.461
Solving for Cv:
0.45 * Cv = 0.461
Cv ≈ 1.025 J/(kg·K)
Now, we can find Cp:
Cp = γ * Cv = 1.45 * 1.025 ≈ 1.484 J/(kg·K)
3. Calculating the Mass of Gas in the Tank
The mass of the gas can be determined 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 ideal gas law to solve for mass (m) gives:
m = (P * V) / (R * T)
Substituting the known values:
m = (900,000 Pa * 0.2 m³) / (0.461 J/(kg·K) * 320 K)
Calculating this yields:
m ≈ 5.57 kg
Summary of Results
To summarize our findings:
- Specific gas constant (R): 0.461 J/(kg·K)
- Specific heat at constant pressure (Cp): 1.484 J/(kg·K)
- Specific heat at constant volume (Cv): 1.025 J/(kg·K)
- Mass of gas in the tank: 5.57 kg
This comprehensive approach allows us to understand the properties of the gas in the tank and how they relate to each other through the ideal gas law and thermodynamic principles.
