--- title: Thermodynamics, Scattered date: 2024-11-10 --- # A Couple of Equations When it comes to energy transfer and phase changes there are only two useful equations. 1. **Specific Heat Equation**: This equation is used to calculate the heat required to change the temperature of a substance without a phase change. $$ \Delta \textbf{KE}=Q = mc\Delta T $$ - $Q$ = heat energy added or removed (in joules, J) - $m$ = mass of the substance (in kilograms, kg) - $c$ = specific heat capacity of the substance (in J/kg°C) - $\Delta T$ = change in temperature (in °C or K) 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. $$ \Delta \textbf{PE} = Q = mL $$ - \( Q \) = heat energy added or removed (in joules, J) - \( m \) = mass of the substance (in kilograms, kg) - \( L \) = latent heat of the substance (in J/kg), which could be the latent heat of fusion (for melting/freezing) or vaporization (for boiling/condensing) These equations together describe the heat energy required for changing the temperature of a substance and for changing its phase. **It is important to note the alternate arrangement of both**: $$ Q=nc_{\mathrm{mol}}\Delta T = \Delta \textbf{KE} $$ or for the other Latent Heat equation: $$ \Delta \textbf{PE} = Q = nL_{\mathrm{mol}} $$ - $n$ is the number of moles - the subscript `mol` notates the same variable, but in units *per* mole. - ==You might see both==, at least they were both on the concept builders. # Black-body radiation 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 $v$). 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. The intensity of the remissions of a black body is given by a function defined so: $$ I(\lambda,T) $$ 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 $\lambda$: $$ I(\lambda, T)\mathrm{d}\lambda $$ 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. ## Wien's displacement law. Wien's law states the wavelength of peak intensity at a given temperature in Kelvin $T$ is given by: $$ \lambda_{\text{peak}}=\frac{b}{T} $$ Where $b$ is Wien's displacement constant, equal to $2.897771955\cdot 10^{-3} \mathrm{m}\cdot\mathrm{K}$. > [!NOTE] > The material does not matter. All black bodies emit radiation over all wavelengths. ## Stefan Boltzmann Law of Radiation This law describes the emissive power of a Black Body per unit area. $$ E_b=\varepsilon \sigma\cdot T^4 $$ In the above equation,: - $E_b$ is the **Emissive Power** of a black body, per unit time, per unit area. - $\varepsilon$ is the emissivity of an object - $\sigma$ is the Boltzmann constant, about $\approx 5.670374419\cdot 10^{-8} \mathrm{W}\cdot\mathrm{m}^{-2}\cdot \mathrm{K}^{-4}$ - A perfect black body has an emissivity of $1$, and an albedo of $0$ - always work in default SI units If you desire the total power across the entirety of the surface: $$ P_b=A\varepsilon \sigma\cdot T^4 $$ Where $A$ is the area of the exposed surface. ## Emissivity and Albedo Emissivity $\varepsilon$ is given by : $$ \varepsilon=\frac{P_{\text{obj}}}{P_b} $$ Albedo is given by: $$ \alpha=\frac{P_\text{reflected}}{P_{\text{incoming}}} $$ 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. Albedo, however, is a far simpler quantity, relying only on the input and output energies of the object. ## Inverse Square Law and Apparent brightness If $b$ is the Power experienced at a radius $r$: $$ b=\frac{L}{4\pi r^2} $$ The above is true when $L$ is procured from the source, most likely using the Boltzmann's law. # Gas Laws and the Ideal Gas Law ## Ideal Gas Law The **Ideal Gas Law** is a fundamental equation describing the behavior of ideal gases: $$ PV = nRT $$ Where: - $P$ = Pressure (in atm, Pa, or another unit) - $V$ = Volume (in liters or cubic meters) - $n$ = Number of moles of gas - $R$ = Ideal gas constant ($0.0821 \, \mathrm{L \cdot atm \cdot mol^{-1} \cdot K^{-1}}$) - $T$ = Temperature (in Kelvin) This equation relates the pressure, volume, temperature, and quantity of an ideal gas. --- ## Boyle's Law **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: $$ P \propto \frac{1}{V} \quad \text{or} \quad P_1V_1 = P_2V_2 $$ ### Key Points: - As volume decreases, pressure increases. - The relationship is hyperbolic. --- ## Charles's Law **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: $$ V \propto T \quad \text{or} \quad \frac{V_1}{T_1} = \frac{V_2}{T_2} $$ ### Key Points: - As temperature increases, volume increases. - Temperature must be in Kelvin. --- ## Gay-Lussac's Law **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: $$ P \propto T \quad \text{or} \quad \frac{P_1}{T_1} = \frac{P_2}{T_2} $$ ### Key Points: - As temperature increases, pressure increases. - Temperature must be in Kelvin. --- ## Avogadro's Law **Avogadro's Law** states that the volume of a gas is directly proportional to the number of moles when pressure and temperature are constant: $$ V \propto n \quad \text{or} \quad \frac{V_1}{n_1} = \frac{V_2}{n_2} $$ ### Key Points: - Adding more gas increases volume proportionally. - This law explains why equal volumes of gases contain equal numbers of molecules at the same temperature and pressure. --- ## Combined Gas Law The **Combined Gas Law** combines Boyle's, Charles's, and Gay-Lussac's Laws into one equation when the number of moles is constant: $$ \frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2} $$ This equation can be used to calculate changes in pressure, volume, or temperature of a gas sample. --- Use these laws to analyze relationships between variables and predict gas behavior under different conditions!