Question

An ideal gas undergoes an expansion from the initial state described by Pi, Vi, T to a final state described by Pf, Vf, T in (a) a process at the constant external pressure Pf, and (b) in a reversible process. Derive expressions for the largest mass that can be lifted through a height h in the surroundings in these processes.

Answer 1

Answer:

A) m = $\frac{p_f(V_f-V_i)}{gh}$

B) m =  $- n R T (ln \frac{V_f }{V_i})\frac{1}{gh}$

Explanation:

A) a process at the constant external pressure Pf,

We Know that

W = mgh-------------------------(1)

W=$p \Delta V$---------------------------(2)

Equating (1) and(2)

mgh =$p \Delta V$

=$p_f (V_f-V_i)$

Therefore m = $\frac{p_f(V_f-V_i)}{gh}$

B)In a irreversible process

W = mgh----------------------(3)

$\mathrm{W}=-\int_{V_{i}}^{V_{f}} \mathrm{P} d V$---------------------(4)

PV =nRT

P =$\frac{nRT}{V}$

$\mathrm{W}=-\int_{V_{i}}^{V_{f}} \frac{\mathrm{nRT}}{V} d V$

W =$-n R T \int_{V_{i}}^{V_{f}} \frac{d V}{V}$

W= $- n R T (ln V_i- ln V_f)$  

W=$- n R T (ln V_f- ln V_i)$

W=$- n R T( ln \frac{V_f }{ V_i})$

From EQ(3)

mgh= $- n R T( ln \frac{V_f }{ V_i})$

m =  $- n R T (ln \frac{V_f }{V_i})\frac{1}{gh}$