Relationship between two specific heat ?

Sol:  dQ = dU + dW; for a perfect gas

dQ at constant pressure

dU at Constant volume; = mC_vdT = mC_v(T₂  - T₁)

dW at constant pressure = PdV = P(V₂  - V₁) = mR(T₂  - T₁) Putting all values we get

dQ = mC_v(T₂  - T₁) + mR(T₂  - T₁)

dQ = m(CV  + R)(T₂  - T₁)

but                      dQ = mC_p(T₂  - T₁)

mC_p(T₂  - T₁) = m(C_V  + R)(T₂  - T₁)

C_p  = C_V  + R; Cp - C_V  = R                                                                                                  ...(i)

Now divided by C_v; we get

C_p /C_V  - 1 = R/C_v; Since C_p /C_V  = y (gama = 1.41)

y - 1 = R/C_v;

or                          C_v  = R/ (y - 1);     C_P  = yR/ (y - 1); C_P>C_V; y>1