How to Derive Mathematical Expressions for the Heat Capactities at Constant Volume and Pressure for an Ideal Gas of Photons

Step-by-Step Guide

1. 1

   Step 1: The concept of heat capacity is derived from the thermodynamic science and is a thermodynamic quantity that is defined as the rate of change of the energy of the given system as a function of temperature.

   There are two variations of heat capacities.
   
   One is defined at constant volume and the other is defined at constant pressure.
   
   Heat capacity is a measure of the given system's ability to absorb heat as a function of the change in temperature.
2. 2

   Step 2: Usually large value of the heat capacity means that the system is able to absorb much heat without significant change in the temperature of the system.

   An example of such a system is the molecule of water which has high value of its heat capacity.
   
   Small value of the heat capacity usually means that a dramatic change in the temperature of the system is associated with little absorption of heat from the surrounding.
   
   In what follows the mathematical expressions of both heat capacities at constant volume and pressure will be derived for an ideal gas of photons. ,, Thus we have:
   Cv=(dp/dT)*c Dp/dT does not have any significant physical value.
   
   Therefore we try to change the differentiation of p from T to t which is the time.
   
   Thus we get:
   Cv=(dp/dt)*(dt/dT)*c Dp/dt is known as the force F1.
   
   Dt/dT is the inverse of the rate of change of temperature as a function of time.
   
   This is still a not convenient expression.
   
   Therefore we change the differentiation from T to r which is the displacement.
   
   Thus we get:
   Cv=(dp/dt)*c/(dT/dt) Cv=(dp/dt)*c/{(dT/dr)*(dr/dt)} This expression has more familiar terms that have physical significance.
   
   Dp/dt is the force F1 ,, This is a well known term in the equation for heat conduction: ,,,,, Thus one has: dQ/dT=Cp=dH/dT=dE+d(PV) Cp=dE/dT+d(PV)/dT ,,,,,, Thus we obtain the following expression for Cp:
   Cp = F1*c*K/{Q*v} + nR
3. 3

   Step 3: We start from the relativistic energy equation of a photon and differentiate it with respect to the temperature T. Thus:

4. 4

   Step 4: E=p*c dE/dT=(dp/dT)*c The first term is nothing but the heat capacity at constant volume Cv.

5. 5

   Step 5: Dr/dt is the expression of the velocity and

6. 6

   Step 6: dT/dr is the rate of change of temperature as a function of the place.

7. 7

   Step 7: Q=k*(dT/dr)

8. 8

   Step 8: Thus the final expression of Cv looks like this: Cv=F1*c/{(Q/K)*v}=F1*c*K/(Q*v)

9. 9

   Step 9: Here Q is the heat that is conducted in the system.

10. 10

    Step 10: Now an expression for Cp or the heat capacity at constant pressure will be derived.

11. 11

    Step 11: At constant pressure we have: Q=H=E+PV Cp is by definition the rate of change of the heat at constant pressure as a function of temperature.

12. 12

    Step 12: For an ideal gas of photons we have: dE/dT=Cv and at constant pressure we have d(PV)=P*dV/dT Thus we obtain: Cp=Cv+PdV/dT

13. 13

    Step 13: We then get an expression for dV/dT from the equation of the ideal gas: PV=nRT Thus we obtain:

14. 14

    Step 14: dV/dT=nR/P

15. 15

    Step 15: Substituting this expression for dV/dT in the above equation for Cp we get:

16. 16

    Step 16: Cp = Cv + nR

17. 17

    Step 17: We remember that: Cv = F1*c*K/(Q*v).

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