Fiveable
🧲AP Physics 2
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🧲AP Physics 2

FRQ 1 – Mathematical Routines
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Unit 9: Thermodynamics
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Guided Practice

Practice FRQ 1 of 121/12
1. A rigid, well-insulated metal container of volume V=2.00×10−2 m3V = 2.00× 10^{-2}\ \text{m}^3V=2.00×10−2 m3 holds n=0.800 moln = 0.800\ \text{mol}n=0.800 mol of helium gas. The gas is initially in thermal equilibrium at temperature T1=300 KT_1 = 300\ \text{K}T1​=300 K and pressure P1P_1P1​. Helium can be treated as an ideal monatomic gas. The container is then placed in thermal contact with a large thermal reservoir at temperature TR=450 KT_R = 450\ \text{K}TR​=450 K by pressing one flat wall of the container against a thick copper slab, as shown in Figure 1. Energy is transferred between the reservoir and the gas only by conduction through the copper slab. The copper slab has thickness L=2.00×10−2 mL = 2.00× 10^{-2}\ \text{m}L=2.00×10−2 m and contact area with the container wall A=5.00×10−3 m2A = 5.00× 10^{-3}\ \text{m}^2A=5.00×10−3 m2. The thermal conductivity of copper is k=400 W m−1 K−1k = 400\ \text{W}\,\text{m}^{-1}\,\text{K}^{-1}k=400 Wm−1K−1. Assume the gas temperature remains uniform throughout the container at all times.

Figure 1. Rigid insulated container of helium pressed against a copper slab connected to a thermal reservoir (heat transfer only by conduction through copper).

A clean, black-and-white physics apparatus diagram with no background clutter. Overall layout (left to right, occupying the full width): - Far left: a large rectangular block labeled exactly "Thermal reservoir" with the text "T_R = 450 K" centered inside the block. - Middle: a vertical rectangular slab between the reservoir and the container, labeled exactly "Copper slab" with the text "k = 400 W·m⁻¹·K⁻¹" printed inside the slab. - Far right: a rigid rectangular metal container (a closed box) labeled exactly "Rigid, sealed metal container". The interior region is labeled exactly "He gas". Contact geometry and dimensions (must be explicitly dimensioned with double-headed arrows): - The copper slab is drawn as a uniform-thickness rectangle whose thickness is measured in the left-to-right direction. - A horizontal double-headed arrow spans ONLY the copper slab thickness (from the reservoir-facing surface of the slab to the container-facing surface of the slab) and is labeled exactly "L = 2.00×10⁻² m". - The area of contact is indicated on the interface between the copper slab and the container wall: show the container’s left wall pressed flush against the slab’s right face. - Place a label with a leader line pointing to the rectangular contact face that reads exactly "Contact area A = 5.00×10⁻³ m²". Container/gas information (printed as visible text adjacent to the container): - Next to the container (outside the box, near its top-right corner), print two lines of text: 1) "V = 2.00×10⁻² m³" 2) "n = 0.800 mol (He)" - Inside the container, include a smaller text line under "He gas": "T increases from 300 K to 450 K". Heat-transfer indication (single unambiguous direction): - Draw ONE thick arrow crossing from the reservoir block, through the copper slab, and pointing into the container wall, all in the left-to-right direction. - Label the arrow directly above it exactly "Energy transfer by conduction". - The arrowhead must point toward the container (rightward), indicating heat flows from the 450 K reservoir to the colder gas. Insulation/rigidity cues (qualitative but explicit): - Around the container’s top, right, and bottom exterior faces (NOT the left face that touches copper), draw a dashed outline labeled exactly "Well-insulated". - Add the word "Rigid" printed on the container wall to emphasize constant volume. No axes, no graph grid. All labels must be clear, horizontal, and fully spelled as given above.

Figure 2. Microscopic motion diagrams for helium at two equilibrium states (State 1 at 300 K, State 2 at 450 K).

A two-panel microscopic motion (vector) diagram, arranged horizontally. Panel layout: - Left panel: a square box labeled above its top edge exactly "State 1". - Right panel: an identical-size square box labeled above its top edge exactly "State 2". - The two boxes are separated by a clear gap equal to roughly one box side length. Temperature labels (must be printed inside each box near the top-left corner): - Inside the State 1 box: "T₁ = 300 K". - Inside the State 2 box: "T₂ = 450 K". Atoms and velocity vectors (must be unambiguous and count-matched): - In EACH box, draw exactly 6 helium atoms as identical small filled circles, distributed so none touch the walls. - From EACH atom, draw one velocity arrow (vector) originating at the atom center. - The set of arrow directions in State 1 and State 2 must be the SAME pattern (to isolate speed as the only difference): for example, arrows pointing in six distinct directions (up, down, left, right, up-right, down-left), with matching directions atom-by-atom between the two panels. Arrow length requirement (numerically constrained by temperature ratio): - All 6 arrows in State 1 must have the same length within that panel. - All 6 arrows in State 2 must have the same length within that panel. - The State 2 arrows must be longer than the State 1 arrows by the factor √(T₂/T₁) = √(450/300) = √(1.5). - To enforce this visually without coordinates: State 1 arrow length is exactly one-half of the box side length divided by 4 (i.e., clearly short and not reaching any wall), and State 2 arrow length is exactly √(1.5) times that State 1 arrow length; ensure the State 2 arrows are noticeably longer but still do not touch any wall. Student prompt indicator: - Below the two panels, centered between them, include the text "Which state has greater average translational kinetic energy?" with a blank line beneath it. Styling constraints: - Atom dots: solid black. - Velocity arrows: solid black, identical thickness in both panels. - Boxes: thin black outlines. - No extra arrows, no motion blur, no random extra dots.

Figure 3. Two microscopic collision-rate panels for helium atoms striking a container wall; compare momentum transfer rate (pressure).

A two-panel diagram arranged horizontally, labeled (a) on the left and (b) on the right. Common structure for BOTH panels: - In each panel, draw a vertical thick line segment near the right side representing a container wall. Label the wall just to the right of the line as "Wall". - The gas region is the space to the left of the wall line. - Draw helium atoms as small filled circles in the gas region. - For each atom that is shown moving toward the wall, draw a rightward velocity arrow pointing toward the wall. - For each depicted collision, show an incoming arrow toward the wall and an outgoing arrow away from the wall (leftward) to represent reversal of momentum. Panel (a): lower pressure depiction via lower collision rate (NOT smaller momentum change per collision): - Place the label "(a)" above the left panel. - Show exactly 4 atoms total in the gas region. - Of these 4 atoms, show exactly 2 atoms undergoing collisions at the wall (each with a paired incoming rightward arrow and outgoing leftward arrow at the wall). - The incoming and outgoing arrow lengths for collisions in panel (a) must be equal (elastic reflection), and must match the arrow lengths used for collisions in panel (b) to ensure the only difference is collision frequency. Panel (b): higher pressure depiction via higher collision rate with the SAME per-collision momentum change: - Place the label "(b)" above the right panel. - Show exactly 8 atoms total in the gas region. - Of these 8 atoms, show exactly 6 atoms undergoing collisions at the wall (each with paired incoming and outgoing arrows at the wall). - The arrow lengths for incoming/outgoing collision arrows must be identical to those in panel (a), indicating equal typical speed in both panels. Momentum-transfer annotation (explicit and comparable): - Under panel (a), print the text "Fewer collisions per unit time". - Under panel (b), print the text "More collisions per unit time". - Between the two panels, centered, add the text: "Greater pressure ↔ greater rate of momentum transfer to the wall". Strict visual consistency requirements: - Wall line thickness is the same in both panels. - Atom dot size is the same in both panels. - All velocity arrows have the same thickness in both panels. - Collision arrows (incoming and outgoing) are drawn at the wall with arrowheads clearly visible and not overlapping the wall label. No numeric axes, no units, no extraneous labels besides those specified.
A.
i. Complete the following tasks in Figures 2 and 3.
• Indicate in Figure 2 whether State 1 or State 2 corresponds to a greater average translational kinetic energy of the helium atoms.
• Indicate in Figure 3 whether panel (a) or panel (b) corresponds to a greater gas pressure on the container wall, based on the rate of momentum transfer from atoms to the wall.
ii. As the container is brought into thermal contact with the reservoir, the helium warms from T1=300 KT_1 = 300\ \text{K}T1​=300 K to T2=450 KT_2 = 450\ \text{K}T2​=450 K while the volume remains constant.
Derive an expression for the final pressure P2P_2P2​ in terms of P1P_1P1​, T1T_1T1​, and T2T_2T2​. Begin your derivation by writing a fundamental physics principle or an equation from the reference information.

Figure 4. Pressure–volume (P–V) diagram for heating the helium at constant volume from T₁ = 300 K to T₂ = 450 K.

A clean P–V graph with a single vertical process line at constant volume. Axes (with exact numeric ranges and tick labels): - Horizontal axis labeled exactly "V (m³)". - Vertical axis labeled exactly "P (Pa)". - The V-axis shows tick labels at: "0", "1.00×10⁻²", "2.00×10⁻²", "3.00×10⁻²". - The P-axis shows symbolic tick labels relative to the initial pressure so the diagram remains valid without computing P₁: tick labels are "0", "P₁", "1.5P₁", "2P₁". - Each axis has an arrowhead at its positive end. State points and process line (exact placement relative to ticks): - Draw a single vertical line located exactly above the V-axis tick labeled "2.00×10⁻²" (this is the constant volume V = 2.00×10⁻² m³). - Mark State 1 as a solid dot on this vertical line exactly at the horizontal level of the tick labeled "P₁". Label it "State 1" to the immediate upper-left of the dot, and next to it print "T₁ = 300 K". - Mark State 2 as a solid dot on the same vertical line exactly at the horizontal level of the tick labeled "1.5P₁". Label it "State 2" to the immediate upper-left of the dot, and next to it print "T₂ = 450 K". - Connect State 1 to State 2 with the vertical line segment (no curve), indicating an isochoric process. Direction indicator: - Add a single arrowhead on the vertical line pointing upward from State 1 toward State 2. - Next to the arrow, print "Heating at constant V". Styling constraints: - Axes: thin black. - Process line: thicker solid black. - State dots: solid black circles. - No gridlines. - No additional curves or shaded regions.
The copper slab remains in contact with the reservoir at TR=450 KT_R = 450\ \text{K}TR​=450 K. While the gas is still at T1=300 KT_1 = 300\ \text{K}T1​=300 K, energy begins to transfer through the copper slab to the gas by conduction. Consider the direction of net energy transfer and the sign of the entropy change of the helium during this warming process.
B. Indicate whether the entropy change of the helium from T1T_1T1​ to T2T_2T2​ is positive, negative, or zero (see Figure 4).
______ Positive
______ Negative
______ Zero
Justify your answer.
Pep