解释一下---**Title:** First Law of Thermodynamics **Mathematical Formulas/Chemical Equations:** \(\Delta E_{int} = Q_{into} + W_{ON}\) \(\Delta E_{int} = Q - W_{BY}\) \(\implies W_{ON} = -W_{BY}\) **Textual Information:** - Where we define \(W\) as the work done ON the gas - The textbook defines \(W\) as the work done BY the gas, i.e. **Annotations/Other Relevant Text:** Total (annotation pointing to \(\Delta E_{int}\) in the first equation) into (subscript on Q in the first equation) ON (subscript on W in the first equation) -HRW (annotation connected by a curved arrow to the definition of W done ON the gas) BY (subscript on W in the second equation) \(= -W_{BY}\) (annotation in the third equation) **Chart/Diagram Description:** * **Type:** Cross-sectional diagram illustrating a system (gas in a cylinder) interacting with its surroundings. * **Main Elements:** * **Cylinder:** A vertical cylinder containing gas particles (represented as blue circles). The inner wall is smooth. * **Piston/Movable Top:** A heavy, movable lid on top of the gas within the cylinder. It is filled with smaller gray spheres labeled "Lead shot". The top surface of the gas and the top surface of the lead shot are slightly concave. * **Outer Jacket:** An outer layer surrounding the cylinder, depicted with a textured pattern, likely representing insulation. * **Base:** A base labeled "Thermal reservoir" is located below the cylinder. * **Control Knob:** A circular object next to the "Thermal reservoir" label, labeled "T" with an arrow pointing to it from a small dot. It is labeled "Control knob". * **Arrows:** * A downward pointing green arrow labeled "W" originates above the piston and points towards the gas, representing work done ON the gas. * A double-headed pink/red arrow labeled "Q" originates from the "Thermal reservoir" and points vertically into the gas, representing heat transfer. * **Labels:** "Lead shot", "W", "Q", "Thermal reservoir", "T", "Control knob". * **Relative Position and Direction:** The piston (with lead shot) is above the gas. The thermal reservoir is below the gas. The work arrow (W) points downwards into the gas. The heat arrow (Q) is double-headed between the thermal reservoir and the gas. **Copyright Information:** \(\copyright\) 2014 John Wiley & Sons, Inc. All rights reserved. **Title:** Work done on a gas **Diagram Description:** * **Left Diagram:** * Type: Cross-sectional illustration of a piston and cylinder system. * Elements: * Cylinder: Vertically oriented, closed at the bottom, open at the top where the piston is located. Appears insulated (textured outer layer). * Piston: Fits inside the cylinder, movable vertically. Shown with "Lead shot" placed on top, indicating a mass or load. * Gas: Occupies the volume below the piston inside the cylinder. * State 1 (Left): Piston is higher, gas volume is labeled Vi (Initial Volume) with purple text. * State 2 (Right): Piston is lower, gas volume is labeled Vf (Final Volume) with purple text. * Movement: A double-headed arrow labeled 'dy' indicates the vertical displacement of the piston between State 1 and State 2. The arrow points downwards next to State 1 and upwards next to State 2, indicating the displacement from the initial to the final position. * Base: Labeled 'To' at the bottom of both states, likely representing the ambient temperature or a reference temperature. * Annotation: "Lead shot" label above the piston surface in both states. A red wavy line on the surface of the lead shot. * **Handwritten Diagram (Right):** * Type: Simplified sketch of a cylinder. * Elements: * Cylinder: Open at the top. * Top Surface/Area: Labeled 'A' (Area) in green. Shaded with green, purple, and red diagonal lines. * Vertical Displacement: Double-headed arrow labeled 'dy' in purple indicates a small vertical segment at the top. * Total Height/Displacement: A green vertical arrow labeled 'h' points downwards from the top surface, indicating a displacement. * Segment Height: A red vertical arrow labeled 'h'' points downwards, indicating a segment of the height below the top surface. * Annotation: Dashed red lines indicate a lower boundary within the cylinder. **Textual Information:** * Piston is pushed down slowly, quasi-statically, so that gas is always in equilibrium: same P and T everywhere, always. * The upward force due to the pressure of the gas: * Incremental work done on gas by pushing piston down by dy: * NB!! Remember dV can be negative! * © 2014 John Wiley & Sons, Inc. All rights reserved. **Mathematical Formulas/Equations:** * `F_gas = P A` * `F_y = -P A_gas` (Note: 'gas' is subscript for A here) * `W = F_y * s` * `dW = F_y dy` * `dW = -P A dy` (Note: 'gas' is subscript for A here) * `= -P dV` (Note: 'f' is subscript for V here, possibly an arrow pointing to dV) **Handwritten Notes:** * `P = F / A` * `dW < 0 Vf < Vi` * `dW_on > 0` (Note: 'on' is subscript for W here) * `dW > 0 Vf > Vi` * `dW < 0` **Title:** Work done on a gas **Diagram Description:** Two diagrams of a cylinder with a piston containing gas. The cylinder appears to be insulated on the sides (indicated by textured pattern). The bottom of the cylinder is labeled with "T₀". The piston holds a container filled with "Lead shot". The left diagram shows the initial state where the piston is higher and the gas occupies a larger volume. There are small dots representing gas particles. The right diagram shows the final state where the piston is lower and the gas occupies a smaller volume. The lead shot level in the piston container is the same in both diagrams. **Description below diagrams:** Piston is pushed down slowly, quasi-statically, so that gas is always in equilibrium: same P and T everywhere, always. **Formula:** W = F_y ⋅ s **Description related to work:** Incremental work done on the gas: dW = -PdV Total work done on gas: W = -∫[from V_i to V_f] PdV **Handwritten Annotation:** P(V) (pointing to the 'P' in the integral formula) **Copyright Notice:** © 2014 John Wiley & Sons, Inc. All rights reserved.