## Heat And Thermodynamics Synopsis

- Heat is a form of energy that exists in matter due to the constant random motion of molecules. Its SI unit is the joule (symbol: J) and cgs unit is the calorie
**(symbol:**cal).1 cal = 4.2 J and 1 kcal = 4200 J. - The temperature of a body indicates its thermal state and determines the direction of the flow of heat Heat always flows from a higher to a lower temperature.
- The absolute zero, or OK, corresponds to a temperature of -273.15 degrees on the Celsius temperature scale. The lower and upper fixed points in the Celsius scale correspond to 273.15 K and 373.15 K respectively.
- The triple point for pure water is 0.01°C (273.16 K) at 611 Pa, and is used to calibrate thermometers.
**Ideal gas equation of state:**pV= nRT, where p = pressure (in pascals or newtons per meter squared), V = volume (in m³), n = amount of substance (in moles), and R = gas constant ≈ 8.3 J K^{-1}mole^{-1}.**van der Waals equation of state for real gases:**\(\left(p+\frac{a}{V^2}\right)\) (V – b) = nRT, where a and b are constants.

**Thermal expansion:**- For length, L
_{2θ}= L_{v}(1 + αθ). - For area, A
_{θ}= A_{0}(1 + βθ). - For volume, V
_{θ}= V_{0}(1 + γθ). - 6α = 3β = 2γ.

- For length, L
- Molar mass M = wNA, where m = mass of each molecule and NA = Avogadro constant ≈ 6.022 x 10
^{23}mol^{-1}. - Boltzmann constant,

\(k=\frac{R}{N_{\mathrm{A}}} \approx \frac{8.3 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}}{6.022 \times 10^{23} \mathrm{~mol}^{-1}} \approx 1.38 \times 10^{-23} \mathrm{~J} \mathrm{~K}^{-1}\) - Gaseous pressure,

\(p=\frac{1}{3}\left(\frac{M}{V}\right) c_{\mathrm{rms}}^2=\frac{1}{3} \rho c_{\mathrm{rms}}^2\).

**In this equation:**- RMS speed, \(c_{\mathrm{rms}}=\sqrt{\frac{3 p}{\rho}}=\sqrt{\frac{3 p V}{M}}=\sqrt{\frac{3 n R T}{M}}=\sqrt{\frac{3 k T}{m}}\), where

m = mass of each molecule; - mean speed, \(\bar{c}=\sqrt{\frac{8 R T}{\pi M}}=\sqrt{\frac{8 k T}{\pi m}}\)

- RMS speed, \(c_{\mathrm{rms}}=\sqrt{\frac{3 p}{\rho}}=\sqrt{\frac{3 p V}{M}}=\sqrt{\frac{3 n R T}{M}}=\sqrt{\frac{3 k T}{m}}\), where
- Most probable velocity, \(c_{\mathrm{mp}}=\sqrt{\frac{2 R T}{M}}=\sqrt{\frac{2 k T}{m}}\)
- Thus, cmp < vmean < crms.
- Heat absorbed (or expelled) Q = mcAT, where Q is in joules, specific heat capacity
**(unit:**J kg^{-1}K^{-1}), and AT = change in temperature (in°C or K). - Specific latent heat = L (
**SI unit:**J kg^{-1}). - During a phase change, Q = mL.
- The molar heat capacity C (
**SI unit:**J mol^{-1}C^{-1}) of gases is process dependent and is given by the equation Q = nCAT, where n = amount of substance (in moles).- At a constant volume, \(C_V=\left.\frac{Q}{n \Delta T}\right|_{V=\text { constant }}\)
- At a constant pressure, \(C_p=\left.\frac{Q}{n \Delta T}\right|_{p=\text { constant }}\)

- \(C_p-C_V=R, \frac{C_p}{C_V}=\gamma, C_p=\left(\frac{R \gamma}{\gamma-1}\right), p V^\gamma=\text { constant }\)
- Change in internal energy, AU = nC
_{v}ΔT. **The first law of thermodynamics:**ΔQ = ΔU + ΔW.**Thermodynamic processes:**

**Molar heat capacity of a mixture of gases:**- \(\left(C_V\right)_{\text {mix }}=\frac{n_1 C_{V_1}+n_2 C_{V_2}+\ldots}{n_1+n_2+\ldots}\)
- \(\left(C_p\right)_{{mix}}=\frac{n_1 C_{p_1}+n_2 C_{p_2}+\ldots}{n_1+n_2+\ldots}\)
- \((\gamma)_{{mix}}=\frac{n_1 C_{p_1}+n_2 C_{p_2}+\ldots}{n_1 C_{V_1}+n_2 C_{V_2}+\ldots}\)

**Equipartition law:**This law states that the average energy of a molecule in a gas associated with each degree of freedom is \(\frac{1}{2}\)kT, where k = Boltzmann constant and T = temperature (in kelvin).