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Question
An ideal gas of density 1.7 × 10−3 g cm−3 at a pressure of 1.5 × 105 Pa is filled in a Kundt's tube. When the gas is resonated at a frequency of 3.0 kHz, nodes are formed at a separation of 6.0 cm. Calculate the molar heat capacities Cp and Cv of the gas.
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Solution
Given:
Density of the ideal gas, ρ = 1.7 × 10−3 g/cm3
= 1.7 k/gm3
Pressure of the gas, P = 1.5 × 105 Pa
R = 8.3 J/mol-K
Resonance frequency of the gas = 3.0 kHz
Node separation in the Kundt's tube
`"l"/2 = 6 "cm"`
So, l = 2×6 = 12 cm = 12 × 10−2 m
So, V = fl = 3 × 103 × 12 × 10−2
= 360 m/s
Speed of sound, V =` sqrt( (gamma"p")/ρ)`
Or `"V"^2 =( gamma"p")/ρ `
`therefore gamma =("v"^2ρ)/"P" = ((360)^2 xx 1.7 xx 10^-3)/(1.5 xx 10^5)`
= 1.4688
Using `"C"_"P" -"C"_"v" = "R" and "C"_"p"/"C"_"v" = gamma`
We know that
`"C"_"v" = "R"/(gamma-1) =8.3/0.4688`
= 17.7 J / mol -K
Cp = R +Cv =8.3 +17.7 = 26 J /mol -K
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