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Thermodynamics

# $\;C_{p}\;$ and $\;C_{v}\;$ denote the molar specific heat capacities of a gas at constant pressure and constant volume respectively . Then

(a) $\;C_{p}-C_{v}\;$ is larger for a diatomic gas than for a monoatomic gas.(b) $\;C_{p}+C_{v}\;$ is larger for a diatomic gas than a monoatomic gas.(c) $\;\large\frac{C_{p}}{C_{v}}\;$ is larger for a diatomic gas than for a monoatomic gas.(d) $\;C_{p}\;.C_{v}\;$ is larger for a diatomic gas than for a monoatomic gas.

Answer : $\;C_{p}+C_{v}\;$ is larger for a diatomic gas than a monoatomic gas.
Explanation :
$C_{v} = \large\frac{f_{2}}{2}R$
$C_{p} = \large\frac{f+2}{f} R$
f is the no . of degrees of freedom.
$\quad\; \qquad$Monoatomic $\qquad \; \qquad \;$ $\qquad \qquad$ Diatomic
f $\quad\; \qquad$ 3 $\qquad \; \qquad \;$ $\qquad \qquad \qquad \qquad$ 5
$C_{v}$ $\quad\; \qquad$ =$\large\frac{3}{2}R$ $\qquad \; \qquad \;$ $\qquad \qquad \qquad \qquad$ $\large\frac{5}{2}R$
$C_{p}$ $\quad\; \qquad$ =$\large\frac{5}{2}R$ $\qquad \; \qquad \;$ $\qquad \qquad \qquad \qquad$ $\large\frac{7}{2}R$
$C_{p} -C_{v}$ $\quad\; \quad$ = R $\qquad \; \qquad \;$ $\qquad \qquad \qquad \qquad$ R
$C_{p} +C_{v}$ $\quad\; \quad$ = 4R $\qquad \; \qquad \;$ $\qquad \qquad \qquad \quad$ 6R
$C_{p}\;.C_{v}$ $\quad\; \qquad$ =$\large\frac{15}{4}R$ $\qquad \; \qquad \;$ $\qquad \quad \qquad \quad$ $\large\frac{25}{4}R$
$\large\frac{C_{p}}{C_{v}}$ $\quad\; \qquad$ =$\large\frac{5}{3}=1.6$ $\qquad \; \qquad \;$ $\qquad \quad \qquad \quad$ $\large\frac{7}{5}=1.4$