202 lines
7.2 KiB
Markdown
202 lines
7.2 KiB
Markdown
---
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title: Thermodynamics, Scattered
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date: 2024-11-10
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---
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# A Couple of Equations
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When it comes to energy transfer and phase changes there are only two useful equations.
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1. **Specific Heat Equation**: This equation is used to calculate the heat required to change the temperature of a substance without a phase change.
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$$
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\Delta \textbf{KE}=Q = mc\Delta T
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$$
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- $Q$ = heat energy added or removed (in joules, J)
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- $m$ = mass of the substance (in kilograms, kg)
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- $c$ = specific heat capacity of the substance (in J/kg°C)
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- $\Delta T$ = change in temperature (in °C or K)
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2. **Latent Heat (Potential Energy Buildup)**: This equation is used for phase changes, where energy changes the state of the substance (like melting or boiling) without changing its temperature.
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$$
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\Delta \textbf{PE} = Q = mL
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$$
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- \( Q \) = heat energy added or removed (in joules, J)
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- \( m \) = mass of the substance (in kilograms, kg)
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- \( L \) = latent heat of the substance (in J/kg), which could be the latent heat of fusion (for melting/freezing) or vaporization (for boiling/condensing)
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These equations together describe the heat energy required for changing the temperature of a substance and for changing its phase.
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**It is important to note the alternate arrangement of both**:
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$$
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Q=nc_{\mathrm{mol}}\Delta T = \Delta \textbf{KE}
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$$
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or for the other Latent Heat equation:
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$$
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\Delta \textbf{PE} = Q = nL_{\mathrm{mol}}
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$$
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- $n$ is the number of moles
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- the subscript `mol` notates the same variable, but in units *per* mole.
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- ==You might see both==, at least they were both on the concept builders.
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# Black-body radiation
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All objects/bodies emit electromagnetic radiation over a range of wavelengths. Cooler objects emit less radiation than warmer bodies, simply because of having less energy. An increase in temperature directly correlates with a smaller wavelength ($\lambda$), and a higher frequency ($f$ or $v$).
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Radiation being *incident* means that it strikes or falls upon a surface, object, or body. Radiation that is incident on an object is partially absorbed and partially reflected. **Thermodynamic Equilibrium** is a state in which a body emits radiation at the same rate that radiation hits it. Therefore, a good absorber of radiation is also a good emitter. A perfect absorber absorbs all E.M. radiation incident on it. Such an object is called a *black body*. This doesn't mean no energy is emitted, however. A black body remits the radiation it absorbs as a thermal radiation in a spectrum determined solely by its temperature.
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The intensity of the remissions of a black body is given by a function defined so:
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$$
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I(\lambda,T)
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$$
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Where $I$ is the *power* intensity that is radiated per unit of wavelength $\lambda$, per unit area of the black body. By multiplying the differential of $\lambda$:
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$$
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I(\lambda, T)\mathrm{d}\lambda
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$$
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Provides the power per unit area of the black body, from the interval $\left[\lambda, \lambda + \mathrm{d}\lambda\right]$, given that $\mathrm{d}\lambda$ is an infinitesimally small quantity.
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## Wien's displacement law.
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Wien's law states the wavelength of peak intensity at a given temperature in Kelvin $T$ is given by:
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$$
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\lambda_{\text{peak}}=\frac{b}{T}
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$$
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Where $b$ is Wien's displacement constant, equal to $2.897771955\cdot 10^{-3} \mathrm{m}\cdot\mathrm{K}$.
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> [!NOTE]
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> The material does not matter. All black bodies emit radiation over all wavelengths.
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## Stefan Boltzmann Law of Radiation
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This law describes the emissive power of a Black Body per unit area.
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$$
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E_b=\varepsilon \sigma\cdot T^4
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$$
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In the above equation,:
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- $E_b$ is the **Emissive Power** of a black body, per unit time, per unit area.
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- $\varepsilon$ is the emissivity of an object
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- $\sigma$ is the Boltzmann constant, about $\approx 5.670374419\cdot 10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot \mathrm{K}^{-4}$
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- A perfect black body has an emissivity of $1$, and an albedo of $0$
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- always work in default SI units
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If you desire the total power across the entirety of the surface:
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$$
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P_b=A\varepsilon \sigma\cdot T^4
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$$
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Where $A$ is the area of the exposed surface.
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## Emissivity and Albedo
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Emissivity $\varepsilon$ is given by :
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$$
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\varepsilon=\frac{P_{\text{obj}}}{P_b}
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$$
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Albedo is given by:
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$$
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\alpha=\frac{P_\text{reflected}}{P_{\text{incoming}}}
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$$
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This is where the Boltzmann law comes in to play. In order to calculate the emissivity of an object, you first need bot: the power of emissions from the target object, and the Power that would be emitted by a black body of the same temperature. This value can be procured using the Stefan Boltzmann Law of Radiation.
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Albedo, however, is a far simpler quantity, relying only on the input and output energies of the object.
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## Inverse Square Law and Apparent brightness
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If $b$ is the Power experienced at a radius $r$:
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$$
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b=\frac{L}{4\pi r^2}
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$$
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The above is true when $L$ is procured from the source, most likely using the Boltzmann's law.
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# Gas Laws and the Ideal Gas Law
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## Ideal Gas Law
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The **Ideal Gas Law** is a fundamental equation describing the behavior of ideal gases:
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$$
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PV = nRT
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$$
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Where:
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- $P$ = Pressure (in atm, Pa, or another unit)
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- $V$ = Volume (in liters or cubic meters)
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- $n$ = Number of moles of gas
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- $R$ = Ideal gas constant ($0.0821 \, \mathrm{L \cdot atm \cdot mol^{-1} \cdot K^{-1}}$)
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- $T$ = Temperature (in Kelvin)
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This equation relates the pressure, volume, temperature, and quantity of an ideal gas.
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---
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## Boyle's Law
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**Boyle's Law** states that the pressure of a gas is inversely proportional to its volume when temperature and the number of moles are constant:
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$$
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P \propto \frac{1}{V} \quad \text{or} \quad P_1V_1 = P_2V_2
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$$
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### Key Points:
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- As volume decreases, pressure increases.
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- The relationship is hyperbolic.
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---
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## Charles's Law
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**Charles's Law** states that the volume of a gas is directly proportional to its absolute temperature when pressure and the number of moles are constant:
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$$
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V \propto T \quad \text{or} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2}
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$$
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### Key Points:
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- As temperature increases, volume increases.
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- Temperature must be in Kelvin.
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---
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## Gay-Lussac's Law
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**Gay-Lussac's Law** states that the pressure of a gas is directly proportional to its absolute temperature when volume and the number of moles are constant:
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$$
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P \propto T \quad \text{or} \quad \frac{P_1}{T_1} = \frac{P_2}{T_2}
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$$
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### Key Points:
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- As temperature increases, pressure increases.
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- Temperature must be in Kelvin.
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---
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## Avogadro's Law
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**Avogadro's Law** states that the volume of a gas is directly proportional to the number of moles when pressure and temperature are constant:
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$$
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V \propto n \quad \text{or} \quad \frac{V_1}{n_1} = \frac{V_2}{n_2}
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$$
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### Key Points:
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- Adding more gas increases volume proportionally.
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- This law explains why equal volumes of gases contain equal numbers of molecules at the same temperature and pressure.
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---
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## Combined Gas Law
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The **Combined Gas Law** combines Boyle's, Charles's, and Gay-Lussac's Laws into one equation when the number of moles is constant:
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$$
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\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}
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$$
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This equation can be used to calculate changes in pressure, volume, or temperature of a gas sample.
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---
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Use these laws to analyze relationships between variables and predict gas behavior under different conditions!
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