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2015-05-18T15:53:37+05:30

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Diatomic ideal gas =>  γ = 1.4
C_v = 5/2 R
C_p = 7/2 R

     C_p = γ C_v

Let us say the mass of the gas in the context to be n moles.   Let the molar mass of the gas be  M.

   P V / T = constant for an ideal gas.

P1, V1, T1      ====>  heated  ===>  P2, V1,  T2
        P2 = 2 P1    =>    T2= 2 T1    as volume is constant.        ---- (1)

2 P1,  V1,  2 T1    ===> heated  ===>  2P1, V3, T3
       V3 = 2 V1    =>   T3 = 2 * (2 T1) = 4 T1      as pressure is constant    --- (2)


During the 1st constant volume heating process:
             ΔQ1  =  n C_v  ΔT        ,  W = 0 as V is constant
                   = n C_v  (2T1 - T1) = n C_v * T1

During the 2nd constant pressure heating process
           ΔQ2  =  n C_p ΔT  =  n C_p * (4T1 - 2T1) = 2 n C_p  T1
                     = 2 n γ C_v T1

Total heat absorbed by the system :
    ΔQ1 + ΔQ2 =     n C_v T1 + 2 n γ C_v T1  =  n T1 (1 + 2 γ) C_v

The total change in the temperature of the system:
       T3 - T1 = ΔT = 4T1 - T1 = 3 T1
      
 Molar Heat capacity of the system
   =   C_m  = Total heat energy supplied / number of moles
   => C_m = [ ΔQ1 + ΔQ2 ] /  [ n ΔT ]
   => C_m =  [ n T1 (1 + 2 γ) C_v  ]  / [ n 3 T1 ]
               =  (1 + 2 γ) C_v / 3

   C_m  = (1 + 2 * 7/5)  5/2 R  / 3
           =  19 / 6 * R

=>  k = 19/6

3 5 3
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