GATE Mechanical Engineering (ME) 2020 Shift 1 Solved Paper

Question : 17 of 65

Marks: +1, -0

For an ideal gas, a constant pressure line and a constant volume line intersect at a point, in the temperature (T) versus specific entropy (s) diagram.

C_{p}

is the specific heat at constant pressure and

{C}_{ {V}}

is the specific heat at constant volume. The ratio of the slopes of the constant pressure and constant volume lines at the point of intersection is

$\;{ {C}_{ {v}}}/{ {C}_{ {p}}}$
$\;{C_{p}}/{C_{v}}$
$\;{C_{p}-C_{v}}/{C_{v}}$
$\;{C_{p}-C_{V}}/{C_{p}}$

Validate

Solution:

Tds equation are
Tds = dU + pdv → (1)
Tds = dU + pdV → (2)
From

1^{\;\text "st "\;}

Tds relation
Tds = dU + pdV
At constant volume, dV = 0
Tds = dU = CvdT

(\;{d T}/{d s})_{V}=\;{T}/{C_{V}}

at constant volume
So, at constant volume, slope of

(T-S)

{T} {/} {Cv}

From

2^{\;\text "nd "\;}

Tds relation,

T {Tds}= {dH}- {vdP}

At constant pressure,

d P=0

T d s=d H=C_{p} d T

(\;{∂ T}/{∂ S})_{p=c}=\;{T}/{C_{p}}

So, ratio of slope of constant pressure & volume

=\;{T}/{C_{p}} \/ \;{T}/{C_{v}}=\;{C_{v}}/{C_{p}}

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