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

A) m =
`(p_f(V_f-V_i))/(gh)`

B) m =
`- n R T (ln (V_f )/(V_i))(1)/(gh)`

Step-by-step 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 =
`(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 =
`(nRT)/(V)`

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

W =
`-n R T \int_{V_(i)}^{V_(f)} (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 (V_f )/( V_i))`

From EQ(3)

mgh=
`- n R T( ln (V_f )/( V_i))`

m =
`- n R T (ln (V_f )/(V_i))(1)/(gh)`