Ncert Exercises & Solutions

Consider that an ideal gas (n moles) is expanding in a process given by p= f(V), which passes through a point ( $P_{0}, V_{0}$ ). Show that the gas is absorbing heat at ( $P_{0}, V_{0}$ ) if the slope of the curve p= f(V) is larger than the slope of the adiabatic passing through ( $P_{0}, V_{0}$ ).

Hint: Use the first law of thermodynamics.

According to the question, the slope of the curve = f(V) where V is volume.

$∴$ The slope of p=f(V) curve at

$(P_{0}, V_{0}) = f (V_{0})$

Step 1: Find the slope of the adiabatic curve.

The slope of adiabatic at

$(P_{0}, V_{0}) = k (- γ) {V_{0}}^{- 1 - γ} = - γ P_{0} / V_{0}$

Step 2: Find the heat absorbed in the process.

Now heat absorbed in the process p = f(V),

$d Q = d U + d W = n C_{v} d T + p d V . . . (i)$

$A s p V = n R T \Rightarrow T = (\frac{1}{n R}) p V$

$\Rightarrow T = (\frac{1}{n R}) V \times f (V)$

$\Rightarrow d T = (\frac{1}{n R}) [f (V) + V \times f' (V)] d V . . . (i i)$

Now from Eq. (i),

$\frac{d Q}{d V} = n C_{v} \frac{d T}{d V} + p \frac{d V}{d V} = n C_{V} \frac{d T}{d V} + p$

$= \frac{n C_{v}}{n R} \times [f (V) + V \times f' (V)] + p$ [from Eq. (ii)]

$= \frac{C_{V}}{R} [f (V) + V \times f' (V)] + f (V)$ [

$∵$ p =f(V)]

$\Rightarrow {[\frac{d Q}{d V}]}_{V = V_{0}} = \frac{C_{V}}{R} [f (V_{0}) + V_{0} \times f' (V_{0})] + f (V_{0})$

$= f (V_{0}) [\frac{C_{V}}{R} + 1] + V_{0} \times f' (V_{0}) \frac{C_{V}}{R}$

$∵ C_{V} = \frac{R}{γ - 1} \Rightarrow \frac{C_{V}}{R} = \frac{1}{γ - 1}$

$\Rightarrow {[\frac{d Q}{d v}]}_{v = v_{0}} = [\frac{1}{γ - 1} + 1] f (V_{0}) + \frac{V_{0} \times f' (V_{0})}{γ - 1}$

$= \frac{γ}{γ - 1} P_{0} + \frac{V_{0}}{γ - 1} \times f' (V_{0})$

Step 3: Find the condition when the heat is absorbed.

Heat is absorbed where

$\frac{d Q}{d V}$ > 0 when the gas expands.

Hence,

$γ P_{0} + V_{0} \times f' (V_{0}) > 0$ or

$f' (V_{0}) > (- γ \frac{p_{o}}{V_{o}})$

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