Thermodynamics - Calorimetry Law

Calorimetry Law

If no heat is wasted to the surroundings then heat gained = heat lost when two bodies are in contact.

Thermal capacity or water equivalent W = mc 

Specific heat of gases is of two type’s specific heat of gases at constant volume (S_v) and specific heat of gases at constant pressure (s_p).

S_v= [ΔQ / m Δ T] constant volume 

And molar specific heat at constant volume

C_v = ( Δ Q / n Δ T) constant volume

s_p = [  Δ Q / m Δ T] constant pressure

And molar specific heat at constant pressure is 

C_p = ( Δ Q / n Δ T) constant pressure

Mayer formula C_p – C_v = R

Note that C_v (adiabatic process) = 0

C_v  (isothermal process) = ∞

Law of equipartition of energy each degree of freedom in a gas average contributes equal energy (average) 1 / 2 KT/ 

Molecule or 1 / 2 RT per mole.

For a monatomic gas

C_v  = 3 / 2 R; C_p = 5 / 2 R and  γ = C_p / C_v = 5 / 3 = 1.67

Number of degrees of freedom = 3 (all translational.).

For the a diatomic gas

C_v  = 5 /2R, C_p = (7/2)R γ= 7/5 = 1.4

Number of degrees of freedom = 5 (3 translation + 2 rotation) assuming molecules if not vibrate.

There are certain gases which do vibrate. In such cases, number of degrees of freedom = 7.

C_v  = ( 7 / 2) R, C_p = (9 / 2) R (C_p / C_v ) = (9 / 2) = 1.29

For polyatomic gases number of degrees of freedom = 6 (3 translation, 2 rotation and 1 vibration)

C_v  = 3R, C_p = 4R γ = C_p / C_v = 4 / 3

If all the degrees of freedom (translational rotational and vibration) are excitable then for an N – atomic molecule (volume or network). Number of degrees of freedom = (6N – 3). For an N – atomic linear molecule number of degrees of freedom = (6N – 5)

Delong petite’s law specific heat of solids at constant volume = 3R or 6 calmol^-10C^-1. It is valid at high temperature. It is based on the fact that there are 3 N vibration states for N molecules. 

For a mixture of gases

γ_mix = C_pmix / C_vmix 

C_vmix = n₁ C_v1 + n² C_v 2 / n₁ + n₂   C_pmix = C_v mix + R 

Specific heat of H₂ is max. [3.5 cal g ^-1°C^-1]. This is followed by water [1 calg ^-1°C^-1] it is minimum for radon and actinium [0.22 calg ^-1 – 0°C^-1].

At low temperature, specific heat C ∝ T₈ (in super-conducting range) and at high temperature C ∝ T. 

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