Heat And Thermodynamics Notes

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 = Lv(1 + αθ).
    • For area, Aθ = A0(1 + βθ).
    • For volume, Vθ = V0(1 + γθ).
    • 6α = 3β = 2γ.
  • Molar mass M = wNA, where m = mass of each molecule and NA = Avogadro constant ≈ 6.022 x 1023 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, \(\bar{c}=\sqrt{\frac{8 R T}{\pi M}}=\sqrt{\frac{8 k T}{\pi m}}\)
  • 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-1K-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-1C-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 = nCvΔT.
  • The first law of thermodynamics: ΔQ = ΔU + ΔW.
  • Thermodynamic processes:

Heat And Thermodynamics 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).

Heat And Thermodynamics Equipartition law

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