JEE MAIN - Chemistry (2019 - 9th January Morning Slot - No. 11)
Consider the reversible isothermal expansion of an ideal gas in a closed system at two different temperatures T1 and T2 (T1 < T2). The correct graphical depiction of the dependence of work done (w) on the final volume (V) is :
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Explanation
Work done in isothermal process,
W = $$-$$ nRT ln$${{{V_2}} \over {{V_1}}}$$
$$ \therefore $$ $$\left| W \right|$$ = nRT ln$${{{V_2}} \over {{V_1}}}$$
$$ \Rightarrow $$ $$\left| W \right|$$ = nRT (lnV2 $$-$$ lnV1)
$$ \Rightarrow $$ $$\left| W \right|$$ = nRT lnV2 $$-$$ nRT lnV1
given that $$V$$2 $$=$$ $$V$$
$$ \therefore $$ $$\left| W \right|$$ = nRT lnV $$-$$ nRT lnV1
By comparing with the straight line equation,
y = mx + C
We get slope is nRT and intercept $$-$$ nRT lnV1 in $$\left| W \right|$$ and lnV graph.
As T2 > T1 So,
Slope nRT2 > nRT1
and intercept
$$-$$ nRT2 lnV1 < $$-$$ nRT1 lnV1
So, we can say
(1) slope of T2 line is more then T1
(2) intercept of T1 line is less negative than T2 line, and intercept of T1 can't be positive, it can be 0 or less than 0 as $$-$$ nRT1lnV1 always $$ \le $$ 0
W = $$-$$ nRT ln$${{{V_2}} \over {{V_1}}}$$
$$ \therefore $$ $$\left| W \right|$$ = nRT ln$${{{V_2}} \over {{V_1}}}$$
$$ \Rightarrow $$ $$\left| W \right|$$ = nRT (lnV2 $$-$$ lnV1)
$$ \Rightarrow $$ $$\left| W \right|$$ = nRT lnV2 $$-$$ nRT lnV1
given that $$V$$2 $$=$$ $$V$$
$$ \therefore $$ $$\left| W \right|$$ = nRT lnV $$-$$ nRT lnV1
By comparing with the straight line equation,
y = mx + C
We get slope is nRT and intercept $$-$$ nRT lnV1 in $$\left| W \right|$$ and lnV graph.
As T2 > T1 So,
Slope nRT2 > nRT1
and intercept
$$-$$ nRT2 lnV1 < $$-$$ nRT1 lnV1
So, we can say
(1) slope of T2 line is more then T1
(2) intercept of T1 line is less negative than T2 line, and intercept of T1 can't be positive, it can be 0 or less than 0 as $$-$$ nRT1lnV1 always $$ \le $$ 0
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