Let's tackle your queries step by step, focusing on the principles of buoyancy, gas laws, and thermodynamics. Each question involves different concepts, so we'll break them down clearly.
1. Buoyant Force and Weight Ratio in a Balloon
To find the ratio of the buoyant force on the balloon to the weight of the air inside it, we can use Archimedes' principle and the ideal gas law.
- Buoyant Force (F_b): The buoyant force is equal to the weight of the displaced air. The volume of the balloon is 1 m³, and the surrounding air is at 300 K.
- Density of Air: At 300 K, the density of air can be approximated using the ideal gas law:
ρ = P / (R * T), where P is atmospheric pressure (approximately 101325 Pa), R is the specific gas constant for air (approximately 287 J/(kg·K)), and T is the temperature in Kelvin.
Thus, ρ = 101325 / (287 * 300) ≈ 1.225 kg/m³.
- Weight of Displaced Air: The weight of the air displaced by the balloon is:
Weight = Volume × Density = 1 m³ × 1.225 kg/m³ = 1.225 kg.
- Weight of Air Inside the Balloon: The air inside the balloon is at 440 K. Using the ideal gas law again, we find the density at 440 K:
ρ_inside = 101325 / (287 * 440) ≈ 0.793 kg/m³.
Therefore, the weight of the air inside the balloon is:
Weight_inside = Volume × Density_inside = 1 m³ × 0.793 kg/m³ = 0.793 kg.
Now, we can find the ratio of the buoyant force to the weight of the air inside:
Ratio = F_b / Weight_inside = 1.225 kg / 0.793 kg ≈ 1.55.
2. Pressure Ratio in a Divided Vessel
For the vessel divided into three compartments, we can apply Dalton's Law of Partial Pressures. Each gas contributes to the total pressure based on its mole fraction.
- Calculate Moles:
- For H₂:
n_H2 = 30 g / 2 g/mol = 15 mol.
- For O₂:
n_O2 = 160 g / 32 g/mol = 5 mol.
- For N₂:
n_N2 = 70 g / 28 g/mol = 2.5 mol.
- Total Moles:
Total n = n_H2 + n_O2 + n_N2 = 15 + 5 + 2.5 = 22.5 mol.
- Pressure Calculation: Using the ideal gas law, the pressure in each compartment can be calculated as:
P = (nRT) / V, where R is the gas constant (0.0821 L·atm/(K·mol)).
- For H₂:
P_H2 = (15 mol × 0.0821 L·atm/(K·mol) × 300 K) / 30 L = 12.31 atm.
- For O₂:
P_O2 = (5 mol × 0.0821 × 300) / 30 = 4.10 atm.
- For N₂:
P_N2 = (2.5 mol × 0.0821 × 300) / 30 = 2.05 atm.
The ratio of pressures in the three compartments is:
Ratio = P_H2 : P_O2 : P_N2 = 12.31 : 4.10 : 2.05.
3. Temperature for Volume Ratio Change in a Piston System
In this scenario, we have a piston dividing a cylinder with two moles of air. Initially, the volume ratio is 4:1, and we want to find the temperature at which this ratio becomes 2:1.
- Initial Conditions: Let the volume of the lower part be V. Then, the upper part is 4V. The total volume is 5V.
- Using Ideal Gas Law: The pressure in both compartments must be equal at equilibrium. Thus, we can set up the equations based on the ideal gas law:
P1V1/T1 = P2V2/T2.
- Final Conditions: When the volume ratio is 2:1, let the lower volume be V' and the upper volume be 2V'. The total volume remains constant, so:
V' + 2V' = 5V, leading to V' = (5/3)V and 2V' = (10/3)V.
- Setting Up the Ratios: The pressures must remain equal, so we can set up the equation:
P1(4V)/T1 = P2(2V)/T2. Since the number of moles and R are constant, we can simplify this to find the relationship between T1 and T2.
By substituting the known values and solving for T2, we can find the temperature at which the volume ratio becomes 2:1. The calculations will yield:
T2 = (T1 * 2) / 4 = 150 K.
In summary, each of these problems illustrates fundamental principles of physics and chemistry, including buoyancy, gas laws, and thermodynamic relationships. If you have further questions or need clarification on any of these points, feel free to ask!
