Match the graphs and the mentioned thermodynamic quantities besides them, in List–I with appropriate integrals in List–II. Assume the system to be 1 mol of an ideal gas.
List–I List–II
I)
P) $| \int y d x |$
II)
Q) $| \int x d y |$
III)
R) $| \int x^{2} d y |$
IV)
S) $| \int \sqrt{y} d x |$
T) $| \int y^{2} d x |$
Choose the correctly matched option:

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$I - R; I I - P; I I I - S; I V - P$

$I - R; I I - P; I I I - T; I V - Q$

$I - T; I I - Q; I I I - R; I V - P$

$I - T; I I - P; I I I - S; I V - Q$

answer is B.

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Detailed Solution

$I - | δ q | = | T d S | y = S, x = T^{\frac{1}{2}} \Rightarrow d S = d y, T = x^{2} \Rightarrow | q | = | \int x^{2} d y | I I - | δ w | = | p d V | y = p, x = V \Rightarrow d V = d x, p = y \Rightarrow | w | = | \int y d x | I I I - | d U | = | c_{v} d T | y = \sqrt{c_{v}}, x = T \Rightarrow d T = d x, c_{v} = y^{2} \Rightarrow | Δ U | = | \int y^{2} d x | I V - | d H | = | c_{p} d T | y = T, x = c_{p} \Rightarrow d T = d y, c_{p} = x \Rightarrow | Δ H | = | \int x d y |$