Important Formulas
Density
$$
\rho = \frac{m}{V}
$$
Combined Gas Law
$$
\frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2}
$$
Pressure Formula
$$
P = \frac{F}{A}
$$
Area of a Circle
$$
A = \pi r²
$$
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1. Answer the questions observing the phase diagram of carbon dioxide (CO₂): (2 points)
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2. A weather balloon rises from ground level to high altitude. At launch, the helium inside the balloon is at 1.0 atm and 22°C, with a volume of 8.0 L. At high altitude, the helium temperature decreases to –10°C, and the balloon's volume increases to 32.0 L. <strong>Find the pressure at that point in scientific notation</strong>: (4 points)
Feedback: Use the combined gas law: \( \frac{P_1 \cdot V_1}{T_1} = \frac{P_2 \cdot V_2}{T_2}\). Convert temperatures to Kelvin: T₁ = 22 + 273 = 295 K, T₂ = -10 + 273 = 263 K. Then: \(P_2 = \frac{P_1 \cdot V_1 \cdot T_2}{V_2 \cdot T_1} = \frac{1.0 \times 8.0 \times 263}{32.0 \times 295} = \frac{2104}{9440} = 0.223 \approx 0.25 \text{ atm} = 2.5 \times 10^{-1} \text{ atm}\)
3. Archimedes attempted to determine whether the king's crown was made of pure gold. To do this, he submerged the crown in a completely filled container, causing it to displace 200 mL of water. Given that the density of gold is 19 000 kg/m³, <strong>what should the mass of the crown be</strong>? (3 points)
Feedback: Using the density formula: \(\rho = \frac{m}{V}\), solve for mass: \(m = \rho \times V\). First convert volume: 200 mL = 0.2 L = 0.0002 m³. Then: \(m = 19,000 \text{ kg/m³} \times 0.0002 \text{ m³} = 3.8 \text{ kg}\). If the crown were pure gold, it should weigh 3.8 kg.
4. A spacecraft collects a sample of neon gas in a rigid, sealed container, so the volume remains constant. Initially, the gas has a pressure of 4.8 × 10⁻² atm and a temperature of 120 K. Later, as the spacecraft enters a different region of space, the gas pressure decreases to 1.2 × 10⁻² atm. <strong>Calculate the final temperature in scientific notation</strong>: (3 points)
Feedback: At constant volume, use Gay-Lussac's Law: \( \frac{P_1}{T_1} = \frac{P_2}{T_2}\). Solve for T₂: \(T_2 = \frac{P_2 \times T_1}{P_1} = \frac{1.2 \times 10^{-2} \times 120}{4.8 \times 10^{-2}} = \frac{1.44}{4.8 \times 10^{-2}} = 30 \text{ K} = 3.0 \times 10^{1} \text{ K}\)
5. Identify the part of the graph where each of the following occurs (refer to a heating curve diagram, e.g., Segment II): (4 points)
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6. Can molecules enter the gas phase even if they haven't reached their boiling point? Explain your answer and provide an everyday example. (2 points)
Feedback: Yes, through evaporation. Surface molecules with sufficient kinetic energy can escape to the gas phase even below the boiling point. Everyday example: wet clothes drying on a line, puddles disappearing after rain, or sweat evaporating from skin even on cool days.
7. Explain what would happen if a liquid solidifies in terms of energy. (2 points)
Feedback: When a liquid solidifies (freezing), it is an exothermic process - energy is released to the surroundings. The molecules lose kinetic energy and organize into a fixed crystalline structure, releasing latent heat.
8. Pressure is a force distributed over an area P=F/A. Suppose an engineer designs a small hydraulic piston that can safely operate at a maximum pressure of 3.5 × 10⁵ Pa (N/m²). The piston has a circular surface with a radius of 3.0 cm. (4 points)
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