Thermodynamics: Proofs of work done on/by gas during adiabatic process

[Matty R]

Hello

I'm really confused with this and would appreciate any help.

Homework Statement

a) Show that the work done on a gas during a quasistatic adiabatic compression is given by:

$$W = \frac{P_f V_f - P_i V_i}{\gamma - 1}$$

b) Show that the work done by a gas during a quasistatic adiabatic compression is given by:

$$w = \frac{P_i V_i - P_f V_f}{\gamma - 1}$$

Homework Equations

$$w = -W$$
$$PV^{\gamma} = K$$

The Attempt at a Solution

a) $$dW = -PdV$$
$$P = KV^{- \gamma}$$
$$W = -K \left[\frac{V^{1- \gamma}}{1 - \gamma} \right]^{V_f} _{V_i}$$
$$W = - \frac{P_f V_f - P_i V_i}{1 - \gamma}$$
$$W = \frac{P_i V_i - P_f V_f}{\gamma - 1}$$

I don't understand where I've gone wrong.

b) $$w = -W$$
$$w = - \frac{P_i V_i - P_f V_f}{\gamma - 1}$$
$$w = \frac{P_f V_f - P_i V_i}{1 - \gamma}$$

I also get this answer if I derive w from:

$$dw = PdV$$

I've probably got a minus sign mixed up, or something like that, but I've been through the derivation so many times it's invading my dreams. Okay, maybe it isn't that bad, but I'm really struggling.

 

[ehild]

Hello

I'm really confused with this and would appreciate any help.

Homework Statement

a) Show that the work done on a gas during a quasistatic adiabatic compression is given by:

$$W = \frac{P_f V_f - P_i V_i}{\gamma - 1}$$


$$w = -W$$
$$PV^{\gamma} = K$$

The Attempt at a Solution

a) $$dW = -PdV$$
$$P = KV^{- \gamma}$$
$$W = -K \left[\frac{V^{1- \gamma}}{1 - \gamma} \right]^{V_f} _{V_i}$$
$$W = - \frac{P_f V_f - P_i V_i}{1 - \gamma}$$
$$W = \frac{P_i V_i - P_f V_f}{\gamma - 1}$$

I don't understand where I've gone wrong.

You have changed sign too many times.
$$W = - \frac{P_f V_f - P_i V_i}{1 - \gamma}=\frac{P_f V_f - P_i V_i}{\gamma-1}$$

 

[Matty R]

Thanks for the reply.

So, erm, is this true:

$$W = - \frac{P_f V_f - P_i V_i}{1 - \gamma}$$
$$= \frac{P_f V_f - P_i V_i}{\gamma-1}$$
$$= \frac{P_i V_i - P_f V_f}{1 - \gamma}$$

 

[ehild]

So, erm, is this true:

$$W = - \frac{P_f V_f - P_i V_i}{1 - \gamma}$$
$$= \frac{P_f V_f - P_i V_i}{\gamma-1}$$
$$= \frac{P_i V_i - P_f V_f}{1 - \gamma}$$

Yes.

ehild

 

[Matty R]

Thank you very much for your help.

I'm feeling quite embarrassed now, though most of the threads I've created here have been about really basic mistakes, so I should be used to it.