Problem 3.3

Given:
Gas is carbon-dioxide,
Case 1: \( T= \) 20^oC = 293 K,
Case 2: \( T= \) 600^oC = 873 K.

To calculate: speed of sound.

For carbon-dioxide (CO₂) the molar mass can be calculated as,

\[ \hat{m}_{CO_{2}}=\text{atomic mass of }C+2\times\text{atomic mass of }O \]

\[ \hat{m}_{CO_{2}}=12.0107+2\times15.999=44.01\,\text{kg/kmol}\ . \]

Therefore, the gas constant is,

\[ R=\frac{R_{u}}{\hat{m}_{CO_{2}}}=\frac{8314}{44.01}=188.9\,\text{J/kg-K}\ . \]

The ratio of specific heats for carbon-dioxide is,

\[ \gamma=1.3\ . \]

The speed of sound can be calculated using the equation,

\[ a=\sqrt{\gamma\,R\,T} \] Case 1: \( T= \) 20^oC = 293 K

\[ \boxed{a=\sqrt{1.3\times188.9\times293}=268.24\,\text{m/s}}\ . \]

Case 2: \( T= \) 600^oC = 873 K

\[ \boxed{a=\sqrt{1.3\times188.9\times873}=463.01\,\text{m/s}}\ . \]

The plot in Fig.1 shows a variation of speed of sound in carbon-dioxide as a function of the temperature.

Fig 1: Speed of sound in carbon-dioxide.