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Homework Statement
Question
If changed isothermal compression process to adiabatic compression process. find the final temperature of process.
Homework Equations
## \alpha = \frac{1}{v} (\frac{∂v}{∂T})_{P} ## expansivity
## \beta = -\frac{1}{v} (\frac{∂v}{∂P})_{T} ## compressibility
##P_1^{1-γ}T_1^{γ}=P_2^{1-γ}T_2^{γ}##
##P_1 = 0.1 Mpa##
##P_2 = 100 Mpa##
##T_1 =15+273.15 K##
##γ= \frac{C_p}{C_v}##
The Attempt at a Solution
I don't know γ . So I must find the γ value.
##Tds = \frac{\beta C_v}{\alpha} dP+\frac{C_p}{\alpha v} d##
In adiabatic process ds = 0 So.
## 0= \frac{\beta C_v}{\alpha} dP_s+\frac{C_p}{\alpha v} dv_s##
## -\frac{\beta C_v}{\alpha} dP_s = \frac{C_p}{\alpha v} dv_s##
## -\frac{\beta \alpha v}{\alpha} dP_s = \frac{C_p}{C_v} dv_s##
So
## γ dv_s =- \beta v dP_s ##
## γ =-\beta v (\frac{dP}{dv})_s ##
I don't have a idea to find γ value . Please help me to find final temperature in a Question.