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Question: \[4.0{\text{ }}g\] of gas occupies \[22.4{\text{ }}L{\text{ }}at{\text{ }}NTP\]. The specific heat c...

4.0 g4.0{\text{ }}g of gas occupies 22.4 L at NTP22.4{\text{ }}L{\text{ }}at{\text{ }}NTP. The specific heat capacity of the gas at constant volume is 5.0 J/K5.0{\text{ }}J/K. If the speed of the sound in this gas at NTP is 952 m/s952{\text{ }}m/s, then the heat capacity at constant pressure is.

(Take gas constant R=8.34 J/K/mol)\left( {Take{\text{ }}gas{\text{ }}constant{\text{ }}R = 8.34{\text{ }}J/K/mol} \right)

A. 8.5JK−1mol−18.5J{K^{ - 1}}mo{l^{ - 1}}
B. 8.0JK−1mol−18.0J{K^{ - 1}}mo{l^{ - 1}}
C. 87.5JK−1mol−187.5J{K^{ - 1}}mo{l^{ - 1}}
D. 7.0JK−1mol−17.0J{K^{ - 1}}mo{l^{ - 1}}

Explanation

Solution

The speed of sound is derived from the kinetic theory of gases in the following way,
v=γRTmv = \sqrt {\dfrac{{\gamma RT}}{m}}

Where, V is the speed, γ\gamma is the adiabatic index, R is the gas constant, T is the temperature and M is the molecular mass.

The adiabatic index is the ratio of specific heat at constant pressure to the specific heat at constant volume.

A gas is said to be containing one mol if it occupies 22.4 L at NTP22.4{\text{ }}L{\text{ }}at {\text{ }}NTP.

Complete step by step solution:

Given at Normal Temperature and Pressure, a 4.0 ga{\text{ }}4.0{\text{ }}ggas occupies 22.4 L22.4{\text{ }}L .

This implies that the moles of gas is 1 mol.(Definition of NTP).\left( {Definition{\text{ }}of{\text{ }}NTP} \right)

Molar mass of the gas m = 4 gm =4×10−3kgm{\text{ }} = {\text{ }}4{\text{ }}gm{\text{ }} = 4 \times {10^{ - 3}}kg

Given that, speed of sound is 952 m/s, T = 273 K952{\text{ }}m/s,{\text{ }}T{\text{ }} = {\text{ }}273{\text{ }}K

Using these in the equation of sound,

v = \sqrt {\dfrac{{\gamma RT}}{m}} \\\ = > {v^2} = \dfrac{{\gamma RT}}{m} \\\ = > {(952)^2} = \dfrac{{\gamma \times 8.3 \times 273}}{{4 \times {{10}^{ - 3}}}} \\\ = > \gamma = \dfrac{{{{(952)}^2} \times 4 \times {{10}^{ - 3}}}}{{8.3 \times 273}} \\\ = > \gamma = 8/5 \\\ \end{gathered} $$ $$\gamma = \dfrac{{{C_p}}}{{{C_v}}}$$ We know that , Given that, the specific heat capacity of the gas at constant volume is $$5.0{\text{ }}J/K/mol$$ I.e., $${C_v} = 5.0J/K/mol$$ Using this value of and $$\gamma = 8/5$$$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;$$ from above, we get, $$\dfrac{8}{5} = \dfrac{{{C_p}}}{5}$$ $${C_p} = 8J{K^{ - 1}}mo{l^{ - 1}}$$ We find that the specific heat at constant pressure is $$8{\text{ }}J/K/mol$$. **Hence the correct option is (B).** **Note:** that specific heat at constant volume$$/$$ pressure has same dimensionality of Gas constant The sound waves are assumed to travel in an adiabatic medium, and not in an isothermal medium as thought by Newton. Which led to the wrong value of Speed of sound. Please keep this in mind.