Question

Calculate the speed of sound in oxygen from the following data. The mass of 22.4 litre of oxygen at STP (T = 273 K and p = 1.0 × 10⁵ N m⁻²) is 32 g, the molar heat capacity of oxygen at constant volume is C_v = 2.5 R and that at constant pressure is C_p = 3.5 R.

Answer 1

Given:
Pressure of oxygen  p = 1.0 × 10⁵ Nm⁻²
Temperature T = 273 K
Mass of oxygen M = 32 g
Volume of oxygen V = 22.4 litre = 22.4\[\times  {10}^{- 3}  m^3\]

Molar heat capacity of oxygen at constant volume C_v = 2.5 R
Molar heat capacity of oxygen at constant pressure C_p = 3.5 R
Density of oxygen \[\rho = \frac{M}{V} = \frac{32  g}{22 . 4 \times {10}^{- 3} m^3}\]

\[We  know  that: \] 

\[\frac{C_p}{C_v} = \gamma\] 

\[ \therefore   \gamma = \frac{3 . 5  R}{2 . 5  R} = 1 . 4\] 

\[\text { Velocity  of  sound  is  given  by: }\] 

\[  v = \sqrt{\frac{\gamma p}{\rho},}\] 

\[\text { where  v  is  the  speed  of  sound . }\] \[\text { On  substituting  the  respective  values  in  the  above  formula,   we  get: }\] 

\[  v = \frac{1 . 4 \times 1 . 0 \times {10}^5}{\left( \frac{32}{22 . 4} \right)}\] 

\[ \Rightarrow v = 310  \text { m/s }\]

Therefore, the speed of sound in oxygen is 310 m/s.