First slide

Different type of processes

Question

4.0 g of a gas occupies 22.4 litres at NTP. The specific heat capacity of the gas at constant volume is 5.0 $J K^{- 1} m o l^{- 1} .$ If the speed of sound in this gas at NTP is 952 $m s^{- 1}$ , then the heat capacity at constant pressure is
(Take gas constant R = 8.3 $J K^{- 1}$ $m o l^{- 1}$ )

Difficult

A

$7.0 {JK}^{- 1} {mol}^{- 1}$
B

$8.5 {JK}^{- 1} {mol}^{- 1}$
C

$8.0 {JK}^{- 1} {mol}^{- 1}$
D

$7.5 {JK}^{- 1} {mol}^{- 1}$

Solution

Since 4.0g of a gas occupies 22.4 litres at NTP, so the molecular mass of the gas is
M=4.0 g $m o l^{- 1}$
As the speed of the sound in the gas is
$v = \sqrt{\frac{γ R T}{M}}$
where $γ$ is the ratio of two specific heats, R is the universal gas constant and T is the temperature of the gas.
$∴ γ = \frac{M v^{2}}{R T}$
$Here, M = 4.0 g {mol}^{- 1} = 4.0 \times 10^{- 3} kg {mol}^{- 1}$
$v = 952 {ms}^{- 1}, R = 8.3 {JK}^{- 1} {mol}^{- 1}$ $and T = 273 K (at NTP)$
$∴ γ = \frac{(4.0 \times 10^{- 3} kg {mol}^{- 1}) {(952 {ms}^{- 1})}^{2}}{(8.3 {JK}^{- 1} {mol}^{- 1}) (273 K)} = 1.6$
By definition,
$γ = \frac{C_{p}}{C_{v}} or C_{p} = γ C_{v}$
$But γ = 1.6 and C_{v} = 5.0 {JK}^{- 1} {mol}^{- 1}$
$∴ C_{p} = (1.6) (5.0 {JK}^{- 1} {mol}^{- 1})$
$= 8.0 {JK}^{- 1} {mol}^{- 1}$

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