🔑:## Step 1: Understanding the Helmholtz Free EnergyThe Helmholtz free energy (F) is a thermodynamic potential that is used to describe the maximum amount of work that can be extracted from a closed system at constant temperature and volume. It is defined as the internal energy (U) of the system minus the product of the temperature (T) and the entropy (S) of the system.## Step 2: Derivation of the Helmholtz Free Energy ExpressionTo derive the expression for Helmholtz free energy, we start with the first law of thermodynamics for a closed system, which states that the change in internal energy (dU) is equal to the heat added to the system (dQ) minus the work done by the system (dW): dU = dQ - dW. For a process at constant temperature and volume, the heat added can be related to the change in entropy (dS) by dQ = TdS.## Step 3: Relating Internal Energy, Entropy, and TemperatureSubstituting dQ = TdS into the first law gives dU = TdS - dW. Rearranging this equation to solve for dW gives dW = dU - TdS. The maximum work that can be done by the system is achieved when the process is reversible, and this maximum work is related to the Helmholtz free energy.## Step 4: Expression for Helmholtz Free EnergyThe Helmholtz free energy (F) is defined such that the change in F (dF) is equal to the change in U minus the change in TS: dF = dU - TdS - SdT. At constant temperature, dT = 0, so dF = dU - TdS. This implies that F = U - TS, since the integration of dF gives F, and the integration constants are absorbed into the definition of F.## Step 5: Implications for Systems at Constant Temperature and VolumeFor systems at constant temperature and volume, the Helmholtz free energy (F = U - TS) represents the maximum amount of work that can be extracted from the system. Subtracting TS from U gives the free energy because it accounts for the energy that is not available to do work due to the entropy of the system. A lower F indicates a greater potential for the system to do work, as more energy is available.The final answer is: boxed{F = U - TS}

🔑:When it comes to cleaning a substrate made of silicon, it's crucial to choose a solvent that effectively removes contaminants without damaging the material. Silicon, being a semiconductor material, has a high reactivity with certain solvents, which can lead to unwanted chemical reactions, etching, or even destruction of the substrate. In this response, we'll explore the properties of silicon, potential reactions with different solvents, and recommend a suitable solvent for cleaning silicon substrates.Properties of Silicon:Silicon is a metalloid with a high affinity for oxygen, which forms a native oxide layer (SiO2) on its surface. This oxide layer is relatively stable and protects the underlying silicon from further oxidation. However, silicon can react with certain solvents, especially those containing hydrogen, fluorine, or chlorine, which can lead to the formation of volatile compounds or etching of the surface.Potential Reactions with Solvents:1. Hydrofluoric acid (HF): Silicon reacts with HF to form silicon tetrafluoride (SiF4) gas, which can etch the surface:Si + 4HF → SiF4 + 2H2This reaction is highly exothermic and can damage the substrate.2. Alkaline solutions (e.g., NaOH, KOH): Silicon can react with alkaline solutions to form silicates, which can lead to etching or pitting:Si + 2NaOH → Na2SiO3 + H23. Chlorinated solvents (e.g., chloroform, dichloromethane): Silicon can react with chlorinated solvents to form silicon chlorides, which can be corrosive:Si + 2CHCl3 → SiCl4 + 2CH3ClRecommended Solvent:Considering the potential reactions mentioned above, a suitable solvent for cleaning silicon substrates is isopropyl alcohol (IPA). IPA is a polar solvent that can effectively remove organic contaminants, such as oils, greases, and residues, without reacting with the silicon substrate.The reasons for choosing IPA are:1. Low reactivity: IPA does not react with silicon or its native oxide layer, minimizing the risk of etching or damage.2. Polarity: IPA is polar, which allows it to effectively dissolve and remove organic contaminants.3. Volatility: IPA has a relatively high vapor pressure, which facilitates easy evaporation and minimizes residue formation.4. Non-corrosive: IPA is non-corrosive and does not contain halogens, which reduces the risk of silicon corrosion or reaction.Additional Considerations:1. Deionized water: While deionized water is not a solvent, it can be used as a final rinse to remove any residual IPA or contaminants.2. Ultrasonic cleaning: Using ultrasonic cleaning with IPA can enhance the cleaning efficiency and remove stubborn contaminants.3. Temperature control: Cleaning should be performed at room temperature or slightly elevated temperatures (up to 40°C) to avoid thermal stress or reaction with the solvent.In conclusion, isopropyl alcohol (IPA) is a suitable solvent for cleaning silicon substrates due to its low reactivity, polarity, volatility, and non-corrosive properties. By choosing the right solvent and following proper cleaning procedures, researchers can effectively remove contaminants from silicon substrates without damaging the material.

🔑:## Step 1: Understanding the ProblemThe problem involves analyzing how the material properties of a microfluidic channel, specifically wall roughness and rigidity, affect the hydrodynamic resistance (Rhyd) within the channel. The channels are made from glass and a flexible polymer, with given dimensions (width = 100 μm, length = 1 cm, depth = 50 μm).## Step 2: Material Properties and Hydrodynamic ResistanceHydrodynamic resistance in microfluidic channels is influenced by the channel's geometry and the properties of the material it's made from, including wall roughness and rigidity. Wall roughness can increase frictional resistance by creating turbulence or altering the flow profile near the walls. Rigidity affects how the channel walls respond to pressure and flow, potentially altering the channel's dimensions during operation.## Step 3: Wall Roughness and Its EffectsGlass channels typically have smoother walls compared to flexible polymers. Smooth walls reduce turbulence and frictional resistance, leading to lower hydrodynamic resistance. In contrast, the rougher surface of polymer channels can increase turbulence and resistance, especially at higher flow rates.## Step 4: Rigidity and Its EffectsThe rigidity of the channel material also plays a crucial role. Glass, being more rigid, maintains its shape under various pressure conditions, ensuring a consistent cross-sectional area for flow. Flexible polymers, however, can deform under pressure, potentially altering the channel's dimensions and thus affecting hydrodynamic resistance.## Step 5: Relevance of Moody's ChartMoody's chart is a tool used to predict friction factors and, by extension, hydrodynamic resistance in channels based on the Reynolds number and relative roughness. For low Reynolds number flows, which are common in microfluidics due to the small scales involved, the flow is typically laminar. Moody's chart can help predict how wall roughness affects the transition from laminar to turbulent flow and the associated increase in resistance.## Step 6: Potential for Turbulence in Flexible ChannelsIn flexible channels, the potential for turbulence at lower Reynolds numbers is higher due to the channel's deformation under flow conditions. This deformation can create irregularities in the flow path, promoting turbulence. Additionally, the rougher surface of flexible materials can further contribute to the onset of turbulence at lower Reynolds numbers compared to smoother, rigid channels like those made of glass.## Step 7: ConclusionThe material properties of microfluidic channels, including wall roughness and rigidity, significantly influence hydrodynamic resistance. Glass channels, with their smooth and rigid characteristics, tend to have lower hydrodynamic resistance compared to channels made from flexible polymers. The use of Moody's chart is relevant for predicting the effects of wall roughness on flow regime transitions, and the potential for turbulence in flexible channels at lower Reynolds numbers must be considered due to their deformable nature and typically higher surface roughness.The final answer is: boxed{R_{hyd, glass} < R_{hyd, polymer}}

🔑:## Step 1: Understanding the ScenarioThe scenario involves a line of observers spaced at standard intervals, all initially at rest relative to each other. As this line extends into regions of the universe that are receding at relativistic speeds, we need to consider the validity of treating the nth observer as being in the same space-time reference system as the first observer.## Step 2: Relativistic Effects and Reference FramesIn special relativity, an inertial reference frame is a frame of reference that is not accelerating. However, as observers move at relativistic speeds relative to each other, their reference frames become non-inertial due to time dilation and length contraction effects. The concept of an extended inertial reference frame becomes problematic at large distances and high speeds because the effects of special relativity, such as time dilation and length contraction, become significant.## Step 3: Implications for Special RelativitySpecial relativity posits that the laws of physics are the same in all inertial frames of reference. However, when considering observers at relativistic speeds relative to each other, the notion of a single, extended inertial reference frame encompassing all observers becomes invalid. Each observer has their own local inertial frame, but these frames are not equivalent due to relativistic effects.## Step 4: General Relativity ConsiderationsGeneral relativity introduces the concept of gravity as the curvature of spacetime caused by mass and energy. While the scenario asks us to disregard gravitational tide and GR field effects, it's essential to note that in reality, these effects would further complicate the definition of a shared reference frame across vast distances. Gravitational effects would cause additional deviations from a flat, Minkowskian spacetime, making the concept of an extended inertial frame even more tenuous.## Step 5: Conclusion on Extended Inertial Reference FramesGiven the relativistic speeds and the vast distances involved, it is not valid to consider the nth observer as being in the same space-time reference system as the first observer. The concept of extended inertial reference frames does not apply in this scenario due to the significant effects of special relativity. Each observer must be considered as having their own local inertial frame, with transformations (Lorentz transformations) necessary to relate measurements between frames.The final answer is: boxed{0}