During an Adiabatic Process, the pressure of a gas is directly proportional to the cube of its absolute temperature. Find the ratio of C_p/Cv. Please help, its important. Thanks!

Dear Student,

Let us begin by defining the meaning of adiabatic process. There is no heat flow from the system to the surroundings or vice-versa in an adiabatic process. 

From 1st law of thermodynamics we know that

dq = dE + dW  ( where dq is the flow of heat which is 0 in this case and dE is the energy and dW is the work done .) 

dW = PdV 

Since force = pressure(P) * area 

and, force * displacement gives work(W) 

for a 3-D gas, work done (W) hence will be = Pressure * dV(volume of container in which the gas expands)

or dW = PdV 

also we know that 

dE/dT = C_v 

hence, dE = C_vdT 

substituting this in the 1st law of thermodynamics we get,

0 = dE + PdV 

0 = C_vdT + PdV 

For real gas PV = RT 

Therefore P = RT/V 

0 = C_vdT + (RT/V)dV 

Dividing throughout by T we get, 

0 = C_vdT/T +(R/V)dV Putting the value of R = C_p - C_v 

0 = C_vdT/T + (C_p – C_v)dV/V

Integrating both sides and taking log we get; 

T^y P^1-y = constant  ( where y is Cp/Cv)

This is the required equation for the given problem. ( the above derivation is for explanation You may use this formula directly to get the result.) 

It is given that pressure (P) is directly proportional to T³ , Putting the value of P as T³ in the above equation we get, 

T^yT^3(1-y) =constant

Now y+3(1-y) = 0 

y+ 3-3y = 0 

3-2y = 0 

3/2 = y. This is the required answer.