warehouse/content/physics/thermodynamics.md

title, date

title	date
Thermodynamics, Scattered	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!