Relationship between two specific heat ?
Sol: dQ = dU + dW; for a perfect gas
dQ at constant pressure
dU at Constant volume; = mC_{v}dT = mC_{v}(T_{2} - T_{1})
dW at constant pressure = PdV = P(V_{2} - V_{1}) = mR(T_{2} - T_{1}) Putting all values we get
dQ = mC_{v}(T_{2} - T_{1}) + mR(T_{2} - T_{1})
dQ = m(CV + R)(T_{2} - T_{1})
but dQ = mC_{p}(T_{2} - T_{1})
mC_{p}(T_{2} - T_{1}) = m(C_{V} + R)(T_{2} - T_{1})
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