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Hi,
I'm thinking about how to derive an expression for the final temperature (call it T2) and the work done (call it W) in an adiabatic compression, when you only know the initial and final pressures (call them P1 and P2), and the initial temperature (call it T1). I'm not sure about my expression though, and would appreciate someone having a look at them.
Since it's an adiabatic process, I infer that
[tex]PV^{K} = const[/tex]
Using the ideal gas law, and rearranging a bit, I make it that
[tex]T_{2} = \frac{P_{1}^{1/K-1}}{P_{2}^{1/K-1}}T_{1}[/tex]
Is that sensible (?).
For the last bit, I ended up deriving the expression
[tex]W = nR\frac{T_{1}-T_{2}}{1-1/K}[/tex]
which to me just looks too simple.
I started with the usual
[tex]W = \int^{V_{1}}_{V_{2}}P dV[/tex]
replacing dV with [tex]-nRT\frac{dP}{P^{2}}[/tex] and the T with [tex]bP^{1-1/K}[/tex] since [tex]b^{1/K-1}T = const. = b \Rightarrow T = bP^{1-1/K}[/tex] to obtain, in the end, the integral:
[tex]W = -nRb \int^{P_{2}}_{P_{1}} P^{-1/K} dP[/tex]
On integrating, I find
[tex]W = nR \frac{bP_{1}^{1-1/K}-bP_{2}^{1-1/K}}{1-1/K}[/tex]
and then substituting for the b's in terms of T1 and P1 or T2 and P2, I found they disappeared, leaving me with
[tex]W = nR\frac{T_{1}-T_{2}}{1-1/K}[/tex]
But I'm not sure that's right (?). How does it look to you?
Cheers!
I'm thinking about how to derive an expression for the final temperature (call it T2) and the work done (call it W) in an adiabatic compression, when you only know the initial and final pressures (call them P1 and P2), and the initial temperature (call it T1). I'm not sure about my expression though, and would appreciate someone having a look at them.
Since it's an adiabatic process, I infer that
[tex]PV^{K} = const[/tex]
Using the ideal gas law, and rearranging a bit, I make it that
[tex]T_{2} = \frac{P_{1}^{1/K-1}}{P_{2}^{1/K-1}}T_{1}[/tex]
Is that sensible (?).
For the last bit, I ended up deriving the expression
[tex]W = nR\frac{T_{1}-T_{2}}{1-1/K}[/tex]
which to me just looks too simple.
I started with the usual
[tex]W = \int^{V_{1}}_{V_{2}}P dV[/tex]
replacing dV with [tex]-nRT\frac{dP}{P^{2}}[/tex] and the T with [tex]bP^{1-1/K}[/tex] since [tex]b^{1/K-1}T = const. = b \Rightarrow T = bP^{1-1/K}[/tex] to obtain, in the end, the integral:
[tex]W = -nRb \int^{P_{2}}_{P_{1}} P^{-1/K} dP[/tex]
On integrating, I find
[tex]W = nR \frac{bP_{1}^{1-1/K}-bP_{2}^{1-1/K}}{1-1/K}[/tex]
and then substituting for the b's in terms of T1 and P1 or T2 and P2, I found they disappeared, leaving me with
[tex]W = nR\frac{T_{1}-T_{2}}{1-1/K}[/tex]
But I'm not sure that's right (?). How does it look to you?
Cheers!