formulae

Temperature Scales and Conversions

• Celsius to Kelvin Conversion: The relationship to find the absolute temperature is T(K)=T(C)+273.15.

• Mechanical Equivalent of Heat: To convert between thermal and mechanical energy units, 1 calorie=4.186 Joules.

Thermal Expansion

• Linear Expansion: The change in length of a material is calculated as ΔL=αL0​ΔT, where α is the coefficient of linear expansion.

• Volume Expansion: For three-dimensional expansion, the formula is ΔV=βV0​ΔT.

• Relationship between Coefficients: For most solid materials, the volume expansion coefficient is approximately three times the linear coefficient, or β=3α.

Heat, Specific Heat, and Calorimetry

• Specific Heat Capacity: The heat (Q) required to change the temperature of a mass (m) is Q=mcΔT.

• Latent Heat (Phase Changes): The energy required to change the phase of a substance without changing its temperature is Q=±mL, where L is the latent heat of fusion or vaporization.

• Conservation of Energy (Calorimetry): In an insulated system, the sum of all heat exchanges must be zero: ∑Q=0.

• Hydrostatic Pressure: In problems involving fluids (such as air bubbles), the total pressure is P=Patm​+ρgh.

Ideal Gas Law and Kinetic Theory

• Ideal Gas Law (Molecular): The state of an ideal gas is defined by PV=NkB​T, where N is the number of molecules and kB​ is Boltzmann’s constant.

• Ideal Gas Law (Molar): Alternatively expressed using moles (n) and the universal gas constant (R) as PV=nRT.

• Boltzmann’s Constant: The value of kB​ is 1.38×10−23 J/K.

• Internal Energy of an Ideal Gas: The total internal energy is proportional to temperature, often expressed as Eint​=23​NkB​T or Eint​=23​nRT.

• Change in Internal Energy: For a temperature change, ΔEint​=23​nRΔT, which can also be written as ΔEint​=23​Δ(PV).

The First Law of Thermodynamics

• The First Law: The conservation of energy for a system is ΔEint​=Q−W, where Q is heat added and W is work done by the system.

• Thermodynamic Work: The infinitesimal work done by an expanding gas is dW=PdV.

• Total Work Done: For a process changing from an initial to a final volume, W=∫Vi​Vf​​PdV, which corresponds to the area under the curve on a P−V diagram.

Formulae for Specific Thermodynamic Processes

• Isochoric (Constant Volume): Since dV=0, the work done is W=0, leading to ΔEint​=Q.

• Isobaric (Constant Pressure): Work is calculated as W=PΔV, and heat is Q=ΔEint​+PΔV.

• Isothermal (Constant Temperature): Since internal energy does not change (ΔEint​=0), the heat added equals the work done: Q=W=nRTln(Vf​/Vi​).

• Adiabatic (No Heat Exchange): Since Q=0, the change in internal energy is the negative of the work done: ΔEint​=−W.

Molar Specific Heats of a Gas

• Heat at Constant Volume: Q=nCv​ΔT.

• Heat at Constant Pressure: Q=nCp​ΔT.

• Mayer’s Relation: The relationship between molar specific heats is Cp​=Cv​+R.

Mathematical Tools

• Taylor Expansion Approximation: For small changes (x≪1), the sources suggest using (1+x)n≈1+nx, specifically in the form 1+x1​≈1−x to simplify complex thermodynamic equations