An ideal gas of density 1.7 × 10^{−3} g cm^{−3} at a pressure of 1.5 × 10^{5} 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 C_{p} and C_{v} of the gas.

#### Solution

Given:

Density of the ideal gas, ρ = 1.7 × 10^{−3} g/cm^{3}

= 1.7 k/gm^{3}

Pressure of the gas, P = 1.5 × 10^{5} 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 × 10^{3} × 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

C_{p} = R +C_{v} =8.3 +17.7 = 26 J /mol -K