---
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.
